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Uniformly Convex and Uniformly Starlike Functions Rosihan M. Ali and V. Ravichandran

arXiv:1106.4377v1 [math.CV] 22 Jun 2011

Abstract A normalized univalent function is uniformly convex if it maps every circular arc contained in the open unit disk with center in it into a convex curve. This article surveys recent results on the class of uniformly convex functions and on an analogous class of uniformly starlike functions.

The long quest for the proof of the conjecture lead to

1. Introduction One of the cornerstones in geometric function theory is the proof of the coefficient conjecture of Bieberbach (1916) by Louis de Branges [13] in the year 1985. The conjecture asserts that the coefficient of a univaP∞ lent function f (z) = z + n=2 an z n in the unit disk D = {z ∈ C : |z| < 1} satisfies |an | ≤ n with strict

inequality unless f is a rotation of the Koebe function z k(z) = . (1 − z)2

In fact, de Branges proved the Milin conjecture (1971) on logarithmic coefficients, which in turn implied the Robertson conjecture (1936) on odd univalent functions, the Rogosinski conjecture (1943) on subordinate functions, and finally the Bieberbach conjecture. Milin’s conjecture asserts that the logarithmic coefficients γn of a P∞ univalent function f (z) = z + n=2 an z n defined by ∞ X f (z) =2 γn z n log z n=1 satisfy the inequality n X 1 (n + 1 − k) k|γk |2 − ≤ 0, k k=1

n = 1, 2, · · · .

The logarithmic coefficients of the Koebe function are γn = 1/n and trivially satisfy the Milin’s conjecture. The Robertson conjecture asserts that the inequality 1 + |c3 |2 + · · · + |c2n−1 |2 ≤ n is satisfied by every odd univalent function of the form g(z) = z + c3 z 3 + c5 z 5 + · · · . Rogosinski conjecture will be stated shortly. The proof that Milin conjecture implies the other conjectures can be found in the books on univalent functions, see for example, [14].

many profound contributions in geometric function theory, particularly the development of various tools for its resolution. These include Loewner’s parametric method, the area method and Grunsky inequalities, Milin’s and FitzGerald’s methods of exponentiating the Grunsky inequalities, Baernstein’s method of maximal functions, and variational methods. Several subclasses of univalent functions were also introduced from geometric considerations and investigated in an attempt to settle the conjecture. Certain subclasses are described below. Let A be the class of all analytic functions in D and normalized by f (0) = 0 = f ′ (0) − 1. Let S be the subclass of A consisting of univalent functions. A domain

D is starlike with respect to a point a ∈ D if every line segment joining the point a to any other point in D lies completely inside D. A domain starlike with respect to the origin is simply called starlike. A domain D is convex if every line segment joining any two points in D lies

completely inside D; in other words, the domain D is convex if and only if it is starlike with respect to every point in D. A function f ∈ S is starlike if f (D) is starlike (with respect to the origin) while it is convex if f (D) is convex. The classes of all starlike and convex functions are respectively denoted by S ∗ and C. Analytically, these classes are characterized by the inequalities ′ zf (z) > 0, f ∈ S ∗ ⇔ Re f (z) and

zf ′′ (z) > 0. f ∈ C ⇔ Re 1 + ′ f (z) More generally, for 0 ≤ α < 1, let S ∗ (α) and C(α) be the subclasses of S consisting of respectively starlike func-

tions of order α, and convex functions of order α. These

2

classes are defined analytically by the inequalities ′ zf (z) > α, f ∈ S ∗ (α) ⇔ Re f (z) and

zf ′′ (z) f ∈ C(α) ⇔ Re 1 + ′ > α. f (z) Another generalization of the class of starlike functions Sγ∗

is the class of strongly starlike functions of order γ, 0 < γ ≤ 1, consisting of f ∈ S satisfying the inequality ′ arg zf (z) < γπ , z ∈ D. f (z) 2

Another related class is the class of close-to-convex functions. A function f ∈ A satisfying the condition ′ f (z) > α, 0 ≤ α < 1, Re g ′ (z)

starlike with respect to w0 if arg(γ(t) − w0 ) is a nondecreasing function of t. The arc γ is convex if the argument of the tangent to γ(t) is a non-decreasing function of t. Definition 1.1 ([16, Definition 1, p. 364], [15, Definition 1, p. 87]). A function f ∈ S is uniformly starlike if f maps every circular arc γ contained in D with center ζ ∈ D onto a starlike arc with respect to f (ζ). The function f ∈ S is uniformly convex if f maps every circular arc γ contained in D with center ζ ∈ D onto a convex arc. The classes of uniformly starlike functions

and uniformly convex functions are denoted respectively by UST and UCV. This article surveys results on uniformly starlike and

for some (not necessarily normalized) convex univalent function g, is called close-to-convex of order α. The

uniformly convex functions. While there is quite a bit of literature on uniformly convex functions, not much is

class of all such functions is denoted by K(α). Closeto-convex functions of order 0 are simply called close-to-

known about uniformly starlike functions. The survey

convex functions. Using the fact that a function f ∈ A with

by Rønning [49] provides a summary of early works on uniformly starlike and uniformly convex functions. 2. Uniformly Starlike Functions

Re(f ′ (z)) > 0 is in S, close-to-convex functions can be shown to be univalent. A function f ∈ A is starlike with respect to symmetric points of order α if 2zf ′ (z) Re > α, f (z) − f (−z)

0 ≤ α < 1.

These functions are also univalent and the class of all such functions is denoted by Ss∗ (α). When α = 0, this class is denoted by Ss∗ . Coefficient estimates for functions

in all these classes can be obtained from the coefficient estimates for functions with positive real part. Starlikeness and convexity are hereditary properties in the sense that every starlike (convex) function maps each disk |z| < r < 1 onto a starlike (convex) domain.

However, Brown [12] showed it is not always true that f ∈ S ∗ maps each disk |z − z0 | < ρ < 1 − |z0 | onto

a domain starlike with respect to f (z0 ). He did prove that the result is true for each f ∈ S and for all sufficiently small disks in D. This motivates the definition of uniformly starlike functions, though it was introduced

independently of the work of Brown [12]. For this purpose, the notion of starlikeness and convexity of curves is needed. Let γ be a curve in D. Then the curve γ is

2.1. Analytic characterization and basic properties. The following two-variable analytic characterization of the class UST is important for obtaining information about functions in the class UST . Theorem 2.1. [16, Theorem 1, p. 365] The function f is in UST if and only if (z − ζ)f ′ (z) ≥ 0, (2.1) Re f (z) − f (ζ)

z, ζ ∈ D.

By taking ζ = −z in the above theorem, evidently

the class UST ⊂ Ss∗ and hence |an | ≤ 1 for f ∈ UST . A better bound |an | ≤ 2/n for f ∈ UST , proved by Charles

Horowitz, was also reported in Goodman [16, Theorem 4, p. 368]. The proof involved showing UST is a subclass

of UST ∗ consisting of functions f ∈ A for which eiα f ′ (z) have positive real part for some real number α. Open Problem 2.1. Determine the sharp coefficient estimates for functions in the class UST of uniformly starlike functions. Using Theorem 2.1, Goodman [16] showed that the function F1 (z) =

1 z ∈ UST ⇔ |A| ≤ √ . 1 − Az 2

Similarly, if F2 (z) = z + Az n , n > 1, and √ 2 , |A| ≤

3

then he showed that F2 is in UST . Merkes and Salamasi [32] improved the bound to be r n+1 |A| ≤ . 2n3 For n 6= 2, the bound need not be sharp. The sharp upper bound was obtained by Nezhmetdinov [33, Corollary 4, p. 47]. The class UST can also be seen to be preserved

under the transformations e−iα f (eiα z) and f (tz)/t, where α ∈ R and 0 < t ≤ 1. For a given locally univalent

analytic function f ∈ A, the disk automorphism is the function Λf : D → C given by Λf (z) :=

f (ϕ(z)) − f (λ) , (1 − |λ|2 )f ′ (λ)

ϕ(z) =

z+λ . 1 + λz

A family F is linearly invariant if Λf ∈ F whenever

f ∈ F . The families S of univalent functions and C of convex functions are linearly invariant families. The disk automorphism of the function F1 with A = 1/2 is not in UST . This shows that the class UST is not a linearly

Open Problem 2.2. Determine the sharp growth, distortion and rotation estimates, as well as the Koebe constant for the class UST . Another application of Theorem 2.1 follows from the simple identity Z 1 ′ f (z) − f (ζ) f (tz + (1 − t)ζ) = dt. (z − ζ)f ′ (z) f ′ (z) 0 Using this identity, Merkes and Salamasi [32, Theorem 4, p. 451] showed that ′ f (w) > 0, z, w ∈ D. f ∈ UST if Re f ′ (z) If f ∈ UST , they also showed that 1/2 ′ f (w) > 0, z, w ∈ D, Re f ′ (z) and the exponent 1/2 is best possible.

invariant family. To provide another application of the above theorem,

2.2. Convolution and Radius Problems. The convolution (or Hadamard product) of two analytic func-

expand the function

tions (z − ζ)f ′ (z) f (z) − f (ζ)

f (z) = z +

Theorem 2.2. [16, Lemma 1, p. 365] Let f ∈ UST ,

and define p0 , p1 , q0 , q1 by

f (ζ)(1 − 2a2 ζ) − ζ , ζ2 f (z) − z q1 (z) = 2 ′ . z f (z)

p1 (z) =

Then |p1 (ζ)| ≤ 2 Re(p0 (ζ)),

and

q1 (z)| ≤ 2 Re(q0 (z)).

Theorem 2.2 and the coefficient estimate |an | ≤ 2/n for f ∈ UST yield the growth inequality for UST : 1 r ≤ |f (z)| ≤ −r + 2 ln , 1 + 2r 1−r

|z| = r < 1.

This inequality provides the lower bound for the Koebe constant for the family UST :

and g(z) = z +

√ 1 3 ≤ K(UST ) ≤ 1 − . 3 4 The upper bound follows from the function f given by √ f (z) = z + 3z 2 /4.

(f ∗ g)(z) := z +

∞ X

bn z n

n=2

is the analytic function

c0 + c1 z + c2 z 2 + · · · with positive real part in D yields the following result:

f (ζ) , ζ f (z) q0 (ζ) = , zf ′ (z)

an z

n

n=2

in its Taylors series in powers of z and ζ respectively. Use of the inequality |cn | ≤ 2 Re c0 for a function p(z) =

p0 (ζ) =

∞ X

∞ X

an b n z n .

n=2

The term “convolution” is used since Z 1 dζ z (f ∗ g)(z) = g(ζ) , f 2πi |ζ|=ρ ζ ζ

|z| < ρ < 1.

The classes of starlike, convex and close-to-convex functions are closed under convolution with convex functions. This was conjectured by P´ olya and Schoenberg [38] and proved by Ruscheweyh and Sheil-Small [58]. Ruscheweyh’s monograph [57] gives a comprehensive survey on convolutions. To make use of this theory in the investigation of the class UST , Merkes and Salamasi [32] proved

the following result.

Theorem 2.3 ([32, Theorem 1, p. 450]). Let f ∈ A. Then f ∈ UST if and only if for all complex numbers α, β with |α| < 1 and |β| < 1, Re

z (1−αz)(1−βz) z f (z) ∗ (1−αz) 2

f (z) ∗

!

≥ 0,

z ∈ D.

The following result of Rønning [51] is also useful in using convolution technique to investigate UST .

4

Theorem 2.4. [51, Lemma 3.3, p. 236] The function f ∈ UST if and only if f (z) − f (xz) ≥ 0, z ∈ D, |x| = 1. (2.2) Re (1 − x)zf ′ (z) Let G denote the subset of A having the property P. If, for every f ∈ F , r−1 f (rz) ∈ G for r ≤ R, and R is

the largest number for which this holds, then R is the G-radius (or the radius of the property P) in F . Thus,

the radius of a property P in the set F is the largest number R such that every function in the set F has the

property P in each disk Dr = {z ∈ D : |z| < r} for every r < R. For example, a starlike function need not be convex; however, every starlike function maps the disk √ |z| < 2 − 3 onto a convex domain and hence the radius √ of convexity of the class S ∗ of starlike functions is 2− 3. Merkes and Salamasi [32] (using Theorem 2.3) and

Rønning [53] (using Theorem 2.4) independently showed that the UST -radius of the class C of convex functions √ is 1/ 2. Merkes and Salamasi [32, Theorem 5, p. 451] also obtained a lower bound for the UST -radius for the class of pre-starlike functions. For α ≤ 1, the class Rα of prestarlike functions of order α consists of functions f ∈ A satisfying z ∗ f ∗ α < 1, (1−z)2−2α ∈ S (α), Re f (z) > 1 , α = 1. z 2

Note that R0 = C and R1/2 = S ∗ (1/2). The known

radius results are recorded in the following theorem. Theorem 2.5.

(1) The UST -radius for the class of univalent functions S is r0 ≈ 0.3691.

(2) The UST -radius r0∗ for the class S ∗ satisfies √ 0.369 < r0∗ ≤ 1/ 7.

(3) The UST -radius for the class of convex func√ tions C is 1/ 2.

(4) The UST -radius for the class of pre-starlike functions is at least (1 + α)/(1 − α) for √ 2−1 √ ≤ α < 1. 2+1

The exact value of the UST -radius r0 of S is obtained as the unique root of ϕ(t) = π/2 in the interval [0, 1] where ϕ(t) is the expression in [53, Equation (2.1), p. 320]. Open Problem 2.3. Determine the (exact) UST -radius r0∗ of the class S ∗ and the exact UST -radius of the class

of pre-starlike functions. Determine whether the class UST is closed under convolution with convex functions. by

For a given subset V ⊂ A, its dual set V ∗ is defined ∗

V :=

(f ∗ g)(z) g∈A: 6 0 for all f ∈ V = z

.

Nezhmetdinov [33, Theorem 2, p. 43] showed that the the dual set of the class UST is the subset of A consisting of functions h : D → C given by z 1 − (w+iα) 1+iα z h(z) = , α ∈ R, |w| = 1. (1 − wz)(1 − z)2 He determined the uniform estimate |an (h)| ≤ dn for the n-th Taylor coefficient of h in the dual set of UST with a √ sharp constant d = M ≈ 1.2557, where M ≈ 1.5770 is the maximum value of a certain trigonometric expression. Using this, he showed that ∞ X

1 n|an | ≤ √ ⇒ f ∈ UST . M n=2 √ The bound 1/ M is sharp. Open Problem 2.4. Rønning [53] proved that UST 6⊂ S ∗ (1/2) and posed the problem of determining the largest α such that UST ⊂ S ∗ (α). Nezhmetdinov [34] showed

that UST 6⊂ S ∗ (α0 ) for some α0 ≈ 0.1483. Determine the largest α such that UST ⊂ S ∗ (α). 3. Uniformly Convex Functions

3.1. Analytic characterizations and parabolic starlike functions. Recall that a univalent function f is in the class UCV of uniformly convex functions if for every circular arc γ contained in D with center ζ ∈ D the image arc f (γ) is convex. From this definition, the following theorem is readily obtained. Theorem 3.1 ([15, Theorem 1, p. 88]). The function f belongs to UCV if and only if f ′′ (z) (3.1) Re 1 + (z − ζ) ′ ≥ 0, f (z)

z, ζ ∈ D.

Though the class C is a linear invariant family, the class UCV is not. This was proved by Goodman [15,

Theorem 5, p. 90] by using the function z . F (z) = 1 − Az

This function F ∈ UCV if and only if |A| ≤ 1/3. From the geometric definition or from Theorem 3.1, it is evident that UCV ⊂ CV. However, by taking ζ = −z

5

in Theorem 3.1, it is evident that UCV ⊂ C(1/2). In view of this inclusion and the coefficient estimate for functions

The class C of convex functions and the class S ∗ of starlike functions are connected by the Alexander result

in C(1/2), the Taylor coefficients an of f ∈ UCV satisfy |an | ≤ 1/n. Unlike the uniformly starlike functions, uni-

that f ∈ C if and only if zf ′ ∈ S ∗ . Such a question between the classes UST and UCV is in fact a question

zation, and this readily yields several important properties of functions in UCV. This one-variable characteriza-

51]) that that there is no inclusion between them:

formly convex functions admit a one-variable characteri-

tion, obtained independently by Rønning [50, Theorem 1, p. 190], and Ma and Minda [29, Theorem 2, p. 162], is the following result. Theorem 3.2. Let f ∈ A. Then f ∈ UCV if and only if ′′ zf (z) zf ′′ (z) , z ∈ D. > ′ (3.2) Re 1 + ′ f (z) f (z)

If f ∈ UCV, then equation (3.2) follows from (3.1) for a suitable choice of ζ. For the converse, the minimum principle for harmonic function is used to restrict ζ and z to |ζ| < |z| < 1. With this restriction, (3.1)

immediately follows from (3.2). To give a nice geometric interpretation of (3.2), let Ωp := {w ∈ C : Re w > |w − 1|}. The set Ωp is the interior of the parabola (Im w)2 = 2 Re w − 1 and it is therefore symmetric with respect to the real axis

and has (1/2, 0) as its vertex. Then f ∈ UCV if and only if zf ′′ (z) ∈ Ωp . 1+ ′ f (z) A class closely related to the class UCV is the class

of equality between UST and SP . It turns out (see [15, UST 6⊂ SP

functions consists of functions f ∈ A satisfying ′ ′ zf (z) zf (z) > − 1 , z ∈ D. Re f (z) f (z)

In other words, the class SP consists of function f = zF ′

where F ∈ UCV.

and

and the sector

The proof follows readily from the implication 1 1 1 ⇒ |w| < = 1 − < 1 − |w| < Re(1 + w). 2 2 2 P∞ A function f (z) = z − n=2 an z n with an ≥ 0 is called a function with negative coefficients. For functions with |w| <

negative coefficients, the above condition is also necessary for a function f to be in UCV or SP (see [11, 61]).

In terms of the coefficients, the results can be stated as follows: Theorem 3.3. Let f be a function of the form f (z) = P∞ z − n=2 an z n with an ≥ 0. Then f ∈ UCV ⇔

and f ∈ SP ⇔

∗ SP ⊂ S ∗ (1/2) ∩ S1/2 .

∞ X

n=2 ∞ X

n(2n − 1)an ≤ 1

(2n − 1)an ≤ 1.

n=2

Denote the class of all functions with negative coefficients by T . Define T UCV := T ∩ UCV, T S ∗ := T ∩ S ∗ ,

T S P := T ∩ SP , and

T C := T ∩ C.

In terms of these classes, the above result can be stated as T UCV = T C(1/2)

and

T S P = T S ∗ (1/2).

For these and other related results, see [11, 61]. Using Theorem 3.3, it can be seen [50] that f (z) = z − An z n ∈ SP ⇔ |An | ≤

{w : | arg w| < π/4}, Rønning [50] noted that

′ zf (z) 1 < ⇒ f ∈ SP . − 1 f (z) 2

(3.4)

Since the parabolic region Ωp is contained in the halfplane {w : Re w > 1/2}

SP 6⊂ UST .

3.2. Examples. To give some examples of functions in UCV and SP , note [40] that ′′ zf (z) 1 (3.3) f ′ (z) < 2 ⇒ f ∈ UCV

of parabolic starlike functions defined below.

Definition 3.1. [50] The class SP of parabolic starlike

and

and f ∈ UCV ⇔ |An | ≤

1 , 2n − 1

1 . n(2n − 1)

6

maps C onto the parabolic region

Goodman [15] showed ∞ X

n=2

∞ X 1 ⇒ f (z) = z + an z n ∈ UCV; 3 n=2

n(n − 1)|an | ≤

this easily follows from Theorem 3.3 since ∞ X

n=2

n(2n − 1)an ≤ 3

∞ X

n=2

n(n − 1)|an | ≤ 1.

The sufficient condition in (3.3) can be extended to a more general circular region. For this purpose, let a >

Ωp := {w ∈ C : Re w > |w − 1|}. Therefore the classes UCV and SP can be expressed in the form zf ′ (z) SP = f ∈ A : ≺ ϕp (z) f (z) and zf ′′ (z) UCV = f ∈ A : 1 + ′ ≺ ϕp (z) . f (z)

1/2. Then it can be shown that the minimum distance from the point w = a to points on the parabola

Rønning [50, Theorem 6, p. 195] went on to show the sharp inequality

|w − 1| = Re w

′

|f (z)| ≤ exp

is given by

Thus [59]

and

a − 1 , if 21 < a ≤ Ra = √ 2 2a − 2, if a ≥ 3 . 2

results are then proved for more general classes of functions by Ma and Minda [30]. For this purpose, let φ be

an analytic function with positive real part in D, nor-

be subordinate to the function F , written f (z) ≺ F (z), if there exists an analytic function w : D → D satisfying w(0) = 0 such that f (z) = F (w(z)). If p : D → C, p(0) = 1 and Re p(z) > 0, then 1+z . 1−z

This follows since the mapping q(z) = (1 + z)/(1 − z) maps D onto the right-half plane ΩH := {w ∈ C : Re w > 0}. In this light, the classes of starlike and convex functions can be expressed as follows: zf ′ (z) 1+z ∗ S = f ∈A: ≺ f (z) 1−z 1+z zf ′′ (z) . ≺ C = f ∈A:1+ ′ f (z) 1−z Rønning [50] and Ma and Minda [29] showed that the function ϕp : D → C defined by

8 =1+ 2 π

log

√ 2 1+ z √ 1− z

2 23 44 4 z + z2 + z3 + z + ··· 3 45 105

malized by the conditions φ(0) = 1 and φ′ (0) > 0, such that φ maps the unit disk D onto a region starlike with respect to 1 that is symmetric with respect to the real axis. They introduced the following classes: zf ′ (z) ∗ ≺ ϕ(z) S (ϕ) := f ∈ A : f (z) and

zf ′′ (z) ≺ ϕ(z) . C(ϕ) = f ∈ A : 1 + ′ f (z) These functions are called Ma-Minda starlike and convex functions respectively. For special choices of ϕ, these classes become well-known classes of starlike and convex functions. For example, for the choice ϕA,B (z) =

and

2 π2

≈ 5.502

gin contained in f (D)) and rotation (the upper bound for | arg(f ′ (z))|) estimates for functions in UCV. These

3.3. Subordination and its consequences. Let f and F be analytic functions in D. Then f is said to

ϕp (z) = 1 +

tortion (bounds for |f ′ (z)|), growth (bounds for |f (z)|), covering (the radius of the largest disk centered at ori-

′ zf (z) < Ra ⇒ f ∈ SP . − a f (z)

(3.5)

14ζ(3) π2

for f ∈ UCV, where ζ(t) denotes the Riemann zeta function. Ma and Minda [29] on the other hand obtained dis-

3 2

′′ 1 + zf (z) − a < Ra ⇒ f ∈ UCV ′ f (z)

p(z) ≺

1 + Az , 1 + Bz

−1 ≤ B < A ≤ 1,

the class S ∗ [A, B] := S ∗ (ϕA,B ) is the class of Janowski starlike functions. For the classes of Ma-Minda starlike and convex functions, the following theorem is obtained.

Theorem 3.4. [30] If f ∈ C(ϕ), then, for |z| = r,

kϕ′ (−r) ≤|f ′ (z)| ≤ kϕ′ (r), −kϕ (−r) ≤ |f (z)| ≤ kϕ (r),

7

where kϕ : D → C is defined by 1+

zkϕ′′ (z) = ϕ(z). kϕ′ (z)

Equality holds for some z 6= 0 if and only if f is a rotation

of kϕ . Also either f is a rotation of kϕ or f (D) contains the disk |w| ≤ −kϕ (−1), where −kϕ (−1) = lim (−kϕ (−r)). r→1−

Further, for |z0 | = r < 1, | arg(f ′ (z0 ))| ≤ max | arg kϕ′ (z)|. |z|=r

The proof relies on the subordination f ′ (z) ≺ kϕ′ (z) satisfied by functions f ∈ C(ϕ). Corresponding results for functions in S ∗ (ϕ) were also obtained by Ma and Minda [30]. The distortion theorem for f ∈ S ∗ (ϕ) re-

quires some additional assumptions on ϕ. Theorem 3.4

contains the corresponding results for uniformly convex functions [29] as special cases. Extension of these (and other closely related) results to functions starlike with respect to symmetric points, conjugate points, multiva-

3.4. Coefficient Problems. As noted earlier, the inclusion UCV ⊂ C(1/2) shows that each Taylor coeffi-

cient an of f ∈ UCV satisfies |an | ≤ 1/n. These bounds can be improved. Since the classes UCV and SP are con-

nected by the Alexander relation that f ∈ UCV if and only if zf ′ ∈ SP , it suffices to give the coefficient estimate for functions in SP .

Theorem 3.6. [50, Theorem 5, p. 194] Let f ∈ SP and P∞ f (z) = z + n=2 an z n . Then (3.6) n c Y c |a2 | ≤ c, and |an | ≤ , 1+ n−1 k−2 k=3

where c = 8/π 2 .

Let p(z) = zf ′ (z)/f (z) = 1 + c1 z + c2 z 2 + · · · , and

p(z) ≺ ϕp (z) where ϕp is given by (3.5). Rogosinski’s

theorem states that |ck | ≤ c for any function p(z) = 1 + c1 z + c2 z 2 + · · · subordinate to the convex univalent

function P (z) = 1 + cz + · · · . The coefficients of f and the coefficients of p are related by

lent starlike functions, and meromorphic functions were investigated in [43, 6, 5]. Let hϕ : D → C be defined by zh′ϕ(z) hϕ (z)

= ϕ(z).

f (z) hϕ (z) ≺ . z z In the case when ϕ is a convex univalent function, this result is a special case of the following general result: f ∈ S ∗ (ϕ) ⇒

Theorem 3.5 (Ruscheweyh [55, Theorem 1, p. 275]). Let φ be a convex function defined in D with φ(0) = 1. Define F by F (z) = z exp

z 0

n−1 X

cn−k ak .

k=1

This together with Rogosinksi’s theorem yield the desired

Ma and Minda [30] proved that

Z

(n − 1)an =

φ(x) − 1 dx . x

The function f belongs to S ∗ (φ) if and only if for all |s| ≤ 1 and |t| ≤ 1, sF (tz) sf (tz) ≺ . tf (sz) tF (sz)

coefficient bounds. Whenever ϕ is a convex univalent function, the bounds for |an | for f ∈ S ∗ (ϕ) is also given

by (3.6) where c := ϕ′ (0). The estimates given by (3.6) are not sharp in general. However, in the case, ϕ(z) = (1+z)/(1−z), the inequalities in (3.6) give sharp bounds for the coefficients of starlike functions.

The sharp coefficient estimates for functions in UCV or SP is still an open problem. However, the sharp esti-

mates of |an | for f ∈ UCV were obtained by Ma and

Minda [29, 31]. They [29, Theorem 5, p. 172] also proved the sharp order of growth |an | = O(1/n2 ) for

f ∈ UCV. The same order of growth holds for f ∈ C(ϕ) if ϕ belongs to the Hardy class of analytic functions H2

(see [30]). They [31] also found the sharp upper bound for the Fekete-Szeg˝ o functional |µa22 − a3 | in the class

UCV for all real µ. For the inverse function f −1 (w) = w +

∞ X

dn wn ,

n=2

Open Problem 3.1. Determine the sharp bound of (n)

they [31] obtained the sharp inequality

∗

|f (z)| for f ∈ C(ϕ) and f ∈ S (ϕ). For f ∈ C(ϕ), the bounds for the cases n = 0, 1 are given by Theorem 3.4. Similar bounds for f ∈ S ∗ (ϕ) are also known with some restrictions on ϕ.

|dn | ≤

8 , (n − 1)nπ 2

n = 2, 3, 4.

More generally, the coefficient problem for f ∈ C(ϕ) is also open.

Estimates for the first two coefficients as

8

well as for the Fekete-Szeg˝ o functional for functions in C(ϕ) were obtained in [30]. For several related coeffi-

3.5. Convolution. Recall that the convolution of two analytic functions

cient problems, see [7].

f (z) = z + Theorem 3.7. Let φ(z) = 1 + B1 z + B2 z 2 + · · · . If 2

3

f (z) = z + a2 z + a3 z + · · · ∈ C(ϕ), then 1 2 2 6 (B2 − (3/2)µB1 + B1 ) if 3B12 µ ≤ 2(B2 + B12 − B1 ) B1 if 2(B2 + B12 − B1 ) ≤ 3B12 µ 6 2 |a3 − µa2 | ≤ ≤ 2(B2 + B12 + B1 ) 1 2 2 6 (−B2 + (3/2)µB1 − B1 ) if 2(B + B 2 + B ) ≤ 3B 2 µ 2 1 1 1 The result is sharp.

To see an outline of the proof, first express the coefficient of f in terms of the coefficients ck for functions with positive real part. For f ∈ C(ϕ), let p : D → C be defined by

p(z) :=

zf ′ (z) = 1 + b1 z + b2 z 2 + · · · f (z)

so that 2a2 = b1 and 6a3 = b2 + b21 . Since φ is univalent and p(z) ≺ φ(z), the function p1 (z) =

1 + φ−1 (p(z)) = 1 + c1 z + c2 z 2 + · · · 1 + φ−1 (p(z))

is analytic and has positive real part in D. Also p1 (z) − 1 (3.7) p(z) = φ p1 (z) + 1 and from this equation (3.7), it follows that 1 1 1 1 B1 c1 and b2 = B1 (c2 − c21 ) + B2 c21 . 2 2 2 4

b1 = Therefore

a3 − µa22 =

(3.8) where v :=

1 2B1

B1 c2 − vc21 , 12

∞ X

an z n ,

and

g(z) = z +

n=2

∞ X

bn z n

n=2

is the analytic function defined by (f ∗ g)(z) := z +

∞ X

an b n z n .

n=2

The convolution of two functions in A is again in A. Since the nth coefficient of normalized univalent function is bounded by n, the convolution of the Koebe function k(z) = z/(1 − z)2 with itself is not univalent. Thus, the

convolution of two univalent (or starlike) functions need not be univalent. P´ olya and Schoenberg [38] conjectured that the class of convex functions C is preserved under

convolution with convex functions:

f, g ∈ C ⇒ f ∗ g ∈ C. In 1973, Ruscheweyh and Sheil-Small [58] (see also [57]) proved the Polya-Schoenberg conjecture. In fact, they also proved that the classes of starlike functions and close-to-convex functions are closed under convolution with convex functions. The proof of these facts follow from the following result which is also used below to show that the classes UCV and SP are closed under convolution with starlike functions of order 1/2. Theorem 3.8. [58, Theorem 2.4, p. 54] Let α ≤ 1, f ∈ Rα and g ∈ S ∗ (α). Then, for any analytic function H ∈ H(D),

f ∗ Hg (D) ⊂ co(H(D)), f ∗g where co(H(D)) denote the closed convex hull of H(U ). Theorem 3.9. [49, Theorem 3.6, p. 131] Let ϕ be a

3 B1 − B12 − B2 + µB12 . 2

convex function with Re ϕ(z) ≥ α, α < 1. If f ∈ Rα and g ∈ S ∗ (ϕ), then f ∗ g ∈ S ∗ (ϕ).

The theorem then follows by an application of the corresponding coefficient results for function with positive real

The proof of this theorem follows readily from Theorem 3.8 by putting H(z) = zg ′ (z)/g(z). In view of

part. Notice that this method is difficult to apply to get bounds for |an | for large n, as an can only be expressed

the fact that f ∈ C(ϕ) if and only if zf ′ ∈ S ∗ (ϕ), an immediate consequence of the above theorem is the cor-

as a non-linear function of the coefficients ck .

Open Problem 3.2. Determine the sharp bound for the Taylor coefficients |an | (n ≥ 5) for f ∈ C(ϕ) and f ∈ ∗

S (ϕ). The same problem for the other classes defined by subordination is still open.

responding result for C(ϕ): if f ∈ Rα and g ∈ C(ϕ), then f ∗ g ∈ C(ϕ) for any convex function ϕ with Re ϕ(z) ≥ α.

In particular, the classes UCV and SP are closed under

convolution with starlike functions of order 1/2. Similar results for several other related classes of functions can be found in [3, 4, 42] or references therein.

9

Goodman remarked that the class UCV is preserved under the transformation e−iα f (eiα z) and no other transformation seems to be available. However, since UCV is closed under convolution with starlike functions of order 1/2 and in particular with convex functions, the following result is obtained.

Γ1 (f (z)) = zf ′ (z), 1 [f (z) + zf ′ (z)] 2 Z k + 1 z k−1 ζ f (ζ)dζ, Re k > 0 Γ3 (f (z)) = zk 0 Z z f (ζ) − f (ηζ) Γ4 (f (z)) = dζ, |η| ≤ 1, η 6= 1. ζ − ηζ 0 Γ2 (f (z)) =

Then Γi (f ) ∈ UCV in |z| < ri whenever f ∈ UCV, where √ 17 − 3 1 ≈ .56155, r3 = r4 = 1. r1 = , r2 = 3 2 3.6. Gaussian Hypergeometric functions. For complex numbers a, b, c ∈ C with c 6= 0, −1, −2, . . . , the

Gaussian hypergeometric function F (a, b; c; z) is defined by the power series ∞ X (a)n (b)n z n . F (a, b, c; z) := (c)n n! n=0

Theorem 3.11. [27, Theorem 4, p. 771] Let a, b ∈ C −

{0} and c > |a| + |b| + 1. If

Γ(c − |a| − |b|)Γ(c) × Γ(c − |a|)Γ(c − |b|) ! 2|ab| 1+ − 1 ≤ 1, c − |a| − |b| − 1

and f ∈ Rη (β), then zF (a, b; c; z) ∗ f (z) ∈ UCV. An extension of these results to other related classes can be found, for example, in [19, 25]. 3.7. Integral transform. The classes UCV and SP

are closed under several integral operators.

Theorem 3.12. [59, Theorem 1, p. 320] Let fi ∈ UCV Pn 1 αi ≤

and αi ’s be real numbers such that αi ≥ 0, and 1. Then the function Z zY n [fi′ (ζ)]αi dζ g(z) = 0 i=1

belongs to UCV.

As an immediate consequence of this theorem, the

Here (a)0 := 1 for a 6= 0 and if n is a positive integer, then (a)n := a(a + 1)(a + 2) · · · (a + n − 1). For β < 1

and η ∈ R, define the class Rη (β) by Rη (β) = f ∈ A | Re(eiη f ′ (z) − β) > 0 for

z∈D .

For the Gaussian hypergeometric function F (a, b, c; z), Kim and Ponnusamy [27] found conditions which would imply that zF (a, b; c; z) belongs to UCV or Rη (β). Further they derived conditions under which f ∈ Rη (β) im-

plies

zF (a, b; c; z) ∗ f (z) ∈ UCV. In fact, by making use of the Gauss summation theorem and Theorem 3.3, they obtained the following sufficient condition for zF (a, b; c; z) ∈ UCV. Theorem 3.10. [27, Theorem 1, p. 768] Let a, b ∈ C − {0} and c > |a| + |b| + 2. If Γ(c − |a| − |b|)Γ(c) × Γ(c − |a|)Γ(c − |b|) 2(|a|)2 (|b|)2 5|ab| + (c − 2 − |a| − |b|)2 c − |a| − |b| − 1 then zF (a, b; c; z) ∈ UCV.

following result provides a mapping of Rη (β) into UCV.

2(1 − β) cos η

Corollary 3.1. [40] Let

1+

They also obtained a weaker condition on the parameters so that the function zF (a, a; c; z) ∈ UCV. The

function g defined by Z g(z) =

0

(0 ≤ αi < 1,

n X 1

n zY 1

(1 − Ai ζ)−2αi dζ

αi ≤ 1, |Ai | ≤

1 , i = 1, 2, . . . , n) 3

belongs to UCV. The first implication in (3.3) yields the

following result.

Theorem 3.13. [59, Theorem 2, p. 320] If f ∈ A satisfies ′ 1 zf (z) f (z) − 1 < 4 , then 2 Z z f (ζ) g(z) = dζ ζ 0 belongs to UCV. 3.8. k-Uniformly convex function. Let k ≥ 0.

A function f ∈ S is called k-uniformly convex in D if the image of every circular arc γ contained in the unit disk

D, with center ζ, |ζ| ≤ k, is convex. For any fixed k ≥ 0,

≤ 2,

the class of all k-uniformly convex functions is denoted by k − UCV. The class k − UCV was introduced and

investigated by Kanas and Wisinowska [22]. As in the

10

case of uniformly convex functions, the following theorem holds. Theorem 3.14 ([22]). Let f ∈ S. Then the following

are equivalent:

arc Γz with center at ζ lying in D is convex α-spirallike. The class of all uniformly convex α-spiral functions is de-

(2) the inequality f ′′ (z) ≥0 Re 1 + (z − ζ) ′ f (z) holds for all z ∈ D and for all |ζ| ≤ k, (3) the inequality ′′ zf (z) zf ′′ (z) Re 1 + ′ > k ′ f (z) f (z) holds for all z ∈ D.

Interestingly, the class of k-uniformly convex functions unifies the class of convex functions (k = 0) and the class of uniformly convex functions (k = 1). Let Ωk = {w : Re w > k|w − 1|}. Then the region Ωk is elliptic for k > 1, parabolic for k = 1, and hyperbolic for 0 < k < 1. The region Ω0 is the right-half plane. Several properties of uniformly convex functions extend to k − UCV functions; these properties are treated in [22, 23, 17, 19, 20, 21]. 3.9. Uniformly spirallike functions. Let Γw be the image of an arc Γz : z = z(t), (a ≤ t ≤ b) under the function f (z) and let w0 be a point not on Γw . Recall that the arc Γw is starlike with respect to w0 if arg(w − w0 ) is a nondecreasing function of t. This condition is equivalent to f ′ (z)z ′ (t) ≥0 f (z) − w0

noted by UCSP(α). An analytic description of UCSP(α) analogous to the class UCV is the following: Theorem 3.16. [46] Let f ∈ A. The the following are equivalent. (1) f ∈ UCSP(α), (2) f satisfies the inequality (z − ζ)f ′′ (z) ≥ 0, Re e−iα 1 + f ′ (z)

z 6= ζ,

(3) f satisfies the inequality ′′ zf (z) zf ′′ (z) , ≥ ′ Re e−iα 1 + ′ f (z) f (z)

z, ζ ∈ D,

z ∈ D.

For f ∈ A, define the function s by iα

f ′ (z) = (s′ (z))e

cos α

.

Then f ∈ UCSP(α) if and only if s ∈ UCV. In view

of this connection with UCV, properties of functions in UCSP can be obtained from the corresponding properties of UCV. The classes of uniformly spirallike and uniformly convex spirallike functions were introduced by

Ravichandran et al. [46], and for a generalization of the class, see [63]. 3.10. Radius problems. The determination of the radius of starlikeness or convexity typically requires an estimate for the real part of the quantities

(a ≤ t ≤ b).

The arc Γw is α-spirallike with respect to w0 if arg

lies between α and α + π. The function f is a uniformly convex α-spiral function if the image of every circular

(1) f ∈ k − UCV,

Im

The arc Γw is convex α-spirallike if ′′ z (t) z ′ (t)f ′′ (z) arg + z ′ (t) f ′ (z)

z ′ (t)f ′ (z) f (z) − w0

QST :=

zf ′ (z) f (z)

and QCV := 1 +

zf ′′ (z) . f ′ (z)

This method of estimating the real part of QST or QCV will not work for the radius problems associated with

lies between α and α + π [14]. The function f is uniformly α-spirallike if the image of every circular arc Γz

uniformly convex functions, parabolic starlike functions, strongly starlike functions and several other subclasses

with center at ζ lying in D is α-spirallike with respect to f (ζ). The class of all uniformly α-spirallike functions is

of starlike/convex functions. In these cases, one need to know the region of values of QST or QCV . This idea was

denoted by USP(α). Here is an analytic description of USP(α) analogous to the class UST . Theorem 3.15. [46] Let |α| <

π 2.

A function f ∈ A

belongs to USP(α) if and only if (z − ζ)f ′ (z) ≥ 0, z 6= ζ, Re e−iα f (z) − f (ζ)

z, ζ ∈ D.

first used by Rønning for computing the sharp radius of parabolic starlikeness for univalent functions. Theorem 3.17. The SP -radius of the class S of univa-

lent functions is 0.33217 and the SP -radius of the class S ∗ of starlike functions is 1/3 ≈ 0.3333 [50, Corollary

3, Theorem 4, p. 192]. The SP -radius of the class C of

11

√ convex functions is 1/ 2 ≈ 0.7071 [51, Theorem 3.1 9b, p. 236].

function f (z) = z + (

Nδ (f ) := The SP -radii for the following classes of functions

were determined by Shanmugam and Ravichandran [59]:

starlike of order α. (2) the class of functions f (z) = z + an z n + · · ·

satisfying the condition Re(f (z)/z) > 0. (3) the class of functions f ∈ A satisfying f (z) g(z) − 1 < 1

f (z) = z + an+1 z n+1 + an+2 z n+2 + · · · satisfying

1 + Az zf ′ (z) ≺ . f (z) 1 + Bz For the special case A = 1 − 2α, B = −1, the class is denoted by Sn∗ (α). Ravichandran, Rønning and ShanSn∗ (β)-radius

mugam [47] investigated and SP -radius for the class Sn∗ [A, B]. They also investigated the radii of Sn∗ (0).

convexity and uniform convexity in Additionally they studied the radius problems for functions whose

an z n ∈ A to be the set

∞ X

k

bk z and

k=2

∞ X

k=2

)

k|ak −bk | ≤ δ .

Ruscheweyh [56] proved among other results that N1/4 (f ) ⊂ S ∗ for f ∈ C. For a more general notion of T -δ-neighbour-

hood of an analytic function, see Sheil-Small and Silvia [60]. Padmanabhan [35] investigated the neighbourhood problem for the class UCV. Since the class UCV is closed under convolution with starlike functions of order 1/2, it follows that the function (f (z) + ǫz)/(1 + ǫ) ∈ SP for |ǫ| < 1/4. Using f ∈ SP ⇔

for some function g starlike of order α.

Rønning [53, Theorem 4, p. 321] showed that the radius of uniform convexity of the classes S and S ∗ is √ (4 − 13)/3 ≈ 0.1314. Let Sn∗ [A, B] consists of functions

n=2

g : g(z) = z+

(1) the class of close-to-starlike functions of order α; these are functions f ∈ A satisfying the condition Re(f (z)/g(z)) > 0 for some function g

P∞

1 (f ∗ h)(z) 6= 0, z

t ∈ R, z ∈ D,

where h(z) :=

2 1 − 2it − t2

z − (1 − z)2

t2 + 1 + it 2

z 1−z

,

Padmanabhan proved that Nδ (f ) ⊂ SP whenever f (z) + ǫz ∈ SP 1+ǫ for |ǫ| < δ < 1. These two assertions together show that N1/8 (f ) ⊂ SP for f ∈ UCV. For some related results, see [17]. References [1] R. Aghalary and S. R. Kulkarni, Certain properties of parabolic starlike and convex functions of order ρ, Bull. Malays. Math. Sci. Soc. (2) 26 (2003), no. 2, 153–162.

derivatives belong to the Kaplan classes K(α, β); their results, in special cases, yield radius results for vari-

[2] R. M. Ali, Starlikeness associated with parabolic regions, Int.

ous classes of close-to-convex functions and functions of

[3] R. M. Ali, A. O. Badghaish and V. Ravichandran, Multivalent

bounded boundary rotation. For 0 ≤ α ≤ β, the Kaplan classes K(α, β) can be defined as follows. A func′

tion f ∈ K(α, β) if and only if there is a function g ∈ S ∗ ((2 + α − β)/2) and a real number t ∈ R such that ′ arg eit zf (z) ≤ απ . g(z) 2

For the radius of uniform convexity of a closely related

class, see [48] wherein they investigated S ∗ (β)-radius and SP -radius of certain integral transforms and Bloch

functions. Related radius results can also be found in [45].

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cheweyh [56] defined the δ-neighbourhood Nδ (f ) of a

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4):265–280, 1997. [48] V. Ravichandran and T. N. Shanmugam. Radius problems for analytic functions. Chinese J. Math., 23(4):343–351, 1995. [49] F. Rønning. A survey on uniformly convex and uniformly starlike functions. Ann. Univ. Mariae Curie-Sklodowska Sect. A, 47:123–134, 1993. [50] F. Rønning. Uniformly convex functions and a corresponding class of starlike functions. Proc. Amer. Math. Soc., 118(1):189–196, 1993.

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[51] F. Rønning. On uniform starlikeness and related properties of univalent functions. Complex Variables Theory Appl., 24(34):233–239, 1994. [52] F. Rønning. Integral representations of bounded starlike functions. Ann. Polon. Math., 60(3):289–297, 1995. [53] F. Rønning. Some radius results for univalent functions. J. Math. Anal. Appl., 194(1):319–327, 1995. [54] T. Rosy, B. A. Stephen, K. G. Subramanian, and H. Silverman. Classes of convex functions. Int. J. Math. Math. Sci., 23(12):819–825, 2000. [55] S. Ruscheweyh, A subordination theorem for F -like functions, J. London Math. Soc. (2) 13 (1976), no. 2, 275–280. [56] St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 521–527. [57] S. Ruscheweyh, Convolutions in Geometric Function Theory, Presses Univ. Montr´ eal, Montreal, Que., 1982. [58] St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the P´ olya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119–135. [59] T. N. Shanmugam and V. Ravichandran. Certain properties of uniformly convex functions. In Computational methods and function theory 1994 (Penang), volume 5 of Ser. Approx. Decompos., pages 319–324. World Sci. Publ., River Edge, NJ, 1995. [60] T. Sheil-Small, E. M. Silvia, Neighborhoods of analytic functions, J. Analyse Math. 52 (1989), 210–240. [61] K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam, and H. Silverman. Subclasses of uniformly convex and uniformly starlike functions. Math. Japon., 42(3):517–522, 1995. [62] K. G. Subramanian, T. V. Sudharsan, P. Balasubrahmanyam, and H. Silverman. Classes of uniformly starlike functions. Publ. Math. Debrecen, 53(3-4):309–315, 1998. [63] N. Xu and D. Yang, On β uniformly convex α-spiral functions, Soochow J. Math. 31 (2005), no. 4, 561–571.

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia E-mail address: [email protected]

Department of Mathematics, University of Delhi, Delhi–110 007, India E-mail address: [email protected]

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arXiv:1106.4377v1 [math.CV] 22 Jun 2011

Abstract A normalized univalent function is uniformly convex if it maps every circular arc contained in the open unit disk with center in it into a convex curve. This article surveys recent results on the class of uniformly convex functions and on an analogous class of uniformly starlike functions.

The long quest for the proof of the conjecture lead to

1. Introduction One of the cornerstones in geometric function theory is the proof of the coefficient conjecture of Bieberbach (1916) by Louis de Branges [13] in the year 1985. The conjecture asserts that the coefficient of a univaP∞ lent function f (z) = z + n=2 an z n in the unit disk D = {z ∈ C : |z| < 1} satisfies |an | ≤ n with strict

inequality unless f is a rotation of the Koebe function z k(z) = . (1 − z)2

In fact, de Branges proved the Milin conjecture (1971) on logarithmic coefficients, which in turn implied the Robertson conjecture (1936) on odd univalent functions, the Rogosinski conjecture (1943) on subordinate functions, and finally the Bieberbach conjecture. Milin’s conjecture asserts that the logarithmic coefficients γn of a P∞ univalent function f (z) = z + n=2 an z n defined by ∞ X f (z) =2 γn z n log z n=1 satisfy the inequality n X 1 (n + 1 − k) k|γk |2 − ≤ 0, k k=1

n = 1, 2, · · · .

The logarithmic coefficients of the Koebe function are γn = 1/n and trivially satisfy the Milin’s conjecture. The Robertson conjecture asserts that the inequality 1 + |c3 |2 + · · · + |c2n−1 |2 ≤ n is satisfied by every odd univalent function of the form g(z) = z + c3 z 3 + c5 z 5 + · · · . Rogosinski conjecture will be stated shortly. The proof that Milin conjecture implies the other conjectures can be found in the books on univalent functions, see for example, [14].

many profound contributions in geometric function theory, particularly the development of various tools for its resolution. These include Loewner’s parametric method, the area method and Grunsky inequalities, Milin’s and FitzGerald’s methods of exponentiating the Grunsky inequalities, Baernstein’s method of maximal functions, and variational methods. Several subclasses of univalent functions were also introduced from geometric considerations and investigated in an attempt to settle the conjecture. Certain subclasses are described below. Let A be the class of all analytic functions in D and normalized by f (0) = 0 = f ′ (0) − 1. Let S be the subclass of A consisting of univalent functions. A domain

D is starlike with respect to a point a ∈ D if every line segment joining the point a to any other point in D lies completely inside D. A domain starlike with respect to the origin is simply called starlike. A domain D is convex if every line segment joining any two points in D lies

completely inside D; in other words, the domain D is convex if and only if it is starlike with respect to every point in D. A function f ∈ S is starlike if f (D) is starlike (with respect to the origin) while it is convex if f (D) is convex. The classes of all starlike and convex functions are respectively denoted by S ∗ and C. Analytically, these classes are characterized by the inequalities ′ zf (z) > 0, f ∈ S ∗ ⇔ Re f (z) and

zf ′′ (z) > 0. f ∈ C ⇔ Re 1 + ′ f (z) More generally, for 0 ≤ α < 1, let S ∗ (α) and C(α) be the subclasses of S consisting of respectively starlike func-

tions of order α, and convex functions of order α. These

2

classes are defined analytically by the inequalities ′ zf (z) > α, f ∈ S ∗ (α) ⇔ Re f (z) and

zf ′′ (z) f ∈ C(α) ⇔ Re 1 + ′ > α. f (z) Another generalization of the class of starlike functions Sγ∗

is the class of strongly starlike functions of order γ, 0 < γ ≤ 1, consisting of f ∈ S satisfying the inequality ′ arg zf (z) < γπ , z ∈ D. f (z) 2

Another related class is the class of close-to-convex functions. A function f ∈ A satisfying the condition ′ f (z) > α, 0 ≤ α < 1, Re g ′ (z)

starlike with respect to w0 if arg(γ(t) − w0 ) is a nondecreasing function of t. The arc γ is convex if the argument of the tangent to γ(t) is a non-decreasing function of t. Definition 1.1 ([16, Definition 1, p. 364], [15, Definition 1, p. 87]). A function f ∈ S is uniformly starlike if f maps every circular arc γ contained in D with center ζ ∈ D onto a starlike arc with respect to f (ζ). The function f ∈ S is uniformly convex if f maps every circular arc γ contained in D with center ζ ∈ D onto a convex arc. The classes of uniformly starlike functions

and uniformly convex functions are denoted respectively by UST and UCV. This article surveys results on uniformly starlike and

for some (not necessarily normalized) convex univalent function g, is called close-to-convex of order α. The

uniformly convex functions. While there is quite a bit of literature on uniformly convex functions, not much is

class of all such functions is denoted by K(α). Closeto-convex functions of order 0 are simply called close-to-

known about uniformly starlike functions. The survey

convex functions. Using the fact that a function f ∈ A with

by Rønning [49] provides a summary of early works on uniformly starlike and uniformly convex functions. 2. Uniformly Starlike Functions

Re(f ′ (z)) > 0 is in S, close-to-convex functions can be shown to be univalent. A function f ∈ A is starlike with respect to symmetric points of order α if 2zf ′ (z) Re > α, f (z) − f (−z)

0 ≤ α < 1.

These functions are also univalent and the class of all such functions is denoted by Ss∗ (α). When α = 0, this class is denoted by Ss∗ . Coefficient estimates for functions

in all these classes can be obtained from the coefficient estimates for functions with positive real part. Starlikeness and convexity are hereditary properties in the sense that every starlike (convex) function maps each disk |z| < r < 1 onto a starlike (convex) domain.

However, Brown [12] showed it is not always true that f ∈ S ∗ maps each disk |z − z0 | < ρ < 1 − |z0 | onto

a domain starlike with respect to f (z0 ). He did prove that the result is true for each f ∈ S and for all sufficiently small disks in D. This motivates the definition of uniformly starlike functions, though it was introduced

independently of the work of Brown [12]. For this purpose, the notion of starlikeness and convexity of curves is needed. Let γ be a curve in D. Then the curve γ is

2.1. Analytic characterization and basic properties. The following two-variable analytic characterization of the class UST is important for obtaining information about functions in the class UST . Theorem 2.1. [16, Theorem 1, p. 365] The function f is in UST if and only if (z − ζ)f ′ (z) ≥ 0, (2.1) Re f (z) − f (ζ)

z, ζ ∈ D.

By taking ζ = −z in the above theorem, evidently

the class UST ⊂ Ss∗ and hence |an | ≤ 1 for f ∈ UST . A better bound |an | ≤ 2/n for f ∈ UST , proved by Charles

Horowitz, was also reported in Goodman [16, Theorem 4, p. 368]. The proof involved showing UST is a subclass

of UST ∗ consisting of functions f ∈ A for which eiα f ′ (z) have positive real part for some real number α. Open Problem 2.1. Determine the sharp coefficient estimates for functions in the class UST of uniformly starlike functions. Using Theorem 2.1, Goodman [16] showed that the function F1 (z) =

1 z ∈ UST ⇔ |A| ≤ √ . 1 − Az 2

Similarly, if F2 (z) = z + Az n , n > 1, and √ 2 , |A| ≤

3

then he showed that F2 is in UST . Merkes and Salamasi [32] improved the bound to be r n+1 |A| ≤ . 2n3 For n 6= 2, the bound need not be sharp. The sharp upper bound was obtained by Nezhmetdinov [33, Corollary 4, p. 47]. The class UST can also be seen to be preserved

under the transformations e−iα f (eiα z) and f (tz)/t, where α ∈ R and 0 < t ≤ 1. For a given locally univalent

analytic function f ∈ A, the disk automorphism is the function Λf : D → C given by Λf (z) :=

f (ϕ(z)) − f (λ) , (1 − |λ|2 )f ′ (λ)

ϕ(z) =

z+λ . 1 + λz

A family F is linearly invariant if Λf ∈ F whenever

f ∈ F . The families S of univalent functions and C of convex functions are linearly invariant families. The disk automorphism of the function F1 with A = 1/2 is not in UST . This shows that the class UST is not a linearly

Open Problem 2.2. Determine the sharp growth, distortion and rotation estimates, as well as the Koebe constant for the class UST . Another application of Theorem 2.1 follows from the simple identity Z 1 ′ f (z) − f (ζ) f (tz + (1 − t)ζ) = dt. (z − ζ)f ′ (z) f ′ (z) 0 Using this identity, Merkes and Salamasi [32, Theorem 4, p. 451] showed that ′ f (w) > 0, z, w ∈ D. f ∈ UST if Re f ′ (z) If f ∈ UST , they also showed that 1/2 ′ f (w) > 0, z, w ∈ D, Re f ′ (z) and the exponent 1/2 is best possible.

invariant family. To provide another application of the above theorem,

2.2. Convolution and Radius Problems. The convolution (or Hadamard product) of two analytic func-

expand the function

tions (z − ζ)f ′ (z) f (z) − f (ζ)

f (z) = z +

Theorem 2.2. [16, Lemma 1, p. 365] Let f ∈ UST ,

and define p0 , p1 , q0 , q1 by

f (ζ)(1 − 2a2 ζ) − ζ , ζ2 f (z) − z q1 (z) = 2 ′ . z f (z)

p1 (z) =

Then |p1 (ζ)| ≤ 2 Re(p0 (ζ)),

and

q1 (z)| ≤ 2 Re(q0 (z)).

Theorem 2.2 and the coefficient estimate |an | ≤ 2/n for f ∈ UST yield the growth inequality for UST : 1 r ≤ |f (z)| ≤ −r + 2 ln , 1 + 2r 1−r

|z| = r < 1.

This inequality provides the lower bound for the Koebe constant for the family UST :

and g(z) = z +

√ 1 3 ≤ K(UST ) ≤ 1 − . 3 4 The upper bound follows from the function f given by √ f (z) = z + 3z 2 /4.

(f ∗ g)(z) := z +

∞ X

bn z n

n=2

is the analytic function

c0 + c1 z + c2 z 2 + · · · with positive real part in D yields the following result:

f (ζ) , ζ f (z) q0 (ζ) = , zf ′ (z)

an z

n

n=2

in its Taylors series in powers of z and ζ respectively. Use of the inequality |cn | ≤ 2 Re c0 for a function p(z) =

p0 (ζ) =

∞ X

∞ X

an b n z n .

n=2

The term “convolution” is used since Z 1 dζ z (f ∗ g)(z) = g(ζ) , f 2πi |ζ|=ρ ζ ζ

|z| < ρ < 1.

The classes of starlike, convex and close-to-convex functions are closed under convolution with convex functions. This was conjectured by P´ olya and Schoenberg [38] and proved by Ruscheweyh and Sheil-Small [58]. Ruscheweyh’s monograph [57] gives a comprehensive survey on convolutions. To make use of this theory in the investigation of the class UST , Merkes and Salamasi [32] proved

the following result.

Theorem 2.3 ([32, Theorem 1, p. 450]). Let f ∈ A. Then f ∈ UST if and only if for all complex numbers α, β with |α| < 1 and |β| < 1, Re

z (1−αz)(1−βz) z f (z) ∗ (1−αz) 2

f (z) ∗

!

≥ 0,

z ∈ D.

The following result of Rønning [51] is also useful in using convolution technique to investigate UST .

4

Theorem 2.4. [51, Lemma 3.3, p. 236] The function f ∈ UST if and only if f (z) − f (xz) ≥ 0, z ∈ D, |x| = 1. (2.2) Re (1 − x)zf ′ (z) Let G denote the subset of A having the property P. If, for every f ∈ F , r−1 f (rz) ∈ G for r ≤ R, and R is

the largest number for which this holds, then R is the G-radius (or the radius of the property P) in F . Thus,

the radius of a property P in the set F is the largest number R such that every function in the set F has the

property P in each disk Dr = {z ∈ D : |z| < r} for every r < R. For example, a starlike function need not be convex; however, every starlike function maps the disk √ |z| < 2 − 3 onto a convex domain and hence the radius √ of convexity of the class S ∗ of starlike functions is 2− 3. Merkes and Salamasi [32] (using Theorem 2.3) and

Rønning [53] (using Theorem 2.4) independently showed that the UST -radius of the class C of convex functions √ is 1/ 2. Merkes and Salamasi [32, Theorem 5, p. 451] also obtained a lower bound for the UST -radius for the class of pre-starlike functions. For α ≤ 1, the class Rα of prestarlike functions of order α consists of functions f ∈ A satisfying z ∗ f ∗ α < 1, (1−z)2−2α ∈ S (α), Re f (z) > 1 , α = 1. z 2

Note that R0 = C and R1/2 = S ∗ (1/2). The known

radius results are recorded in the following theorem. Theorem 2.5.

(1) The UST -radius for the class of univalent functions S is r0 ≈ 0.3691.

(2) The UST -radius r0∗ for the class S ∗ satisfies √ 0.369 < r0∗ ≤ 1/ 7.

(3) The UST -radius for the class of convex func√ tions C is 1/ 2.

(4) The UST -radius for the class of pre-starlike functions is at least (1 + α)/(1 − α) for √ 2−1 √ ≤ α < 1. 2+1

The exact value of the UST -radius r0 of S is obtained as the unique root of ϕ(t) = π/2 in the interval [0, 1] where ϕ(t) is the expression in [53, Equation (2.1), p. 320]. Open Problem 2.3. Determine the (exact) UST -radius r0∗ of the class S ∗ and the exact UST -radius of the class

of pre-starlike functions. Determine whether the class UST is closed under convolution with convex functions. by

For a given subset V ⊂ A, its dual set V ∗ is defined ∗

V :=

(f ∗ g)(z) g∈A: 6 0 for all f ∈ V = z

.

Nezhmetdinov [33, Theorem 2, p. 43] showed that the the dual set of the class UST is the subset of A consisting of functions h : D → C given by z 1 − (w+iα) 1+iα z h(z) = , α ∈ R, |w| = 1. (1 − wz)(1 − z)2 He determined the uniform estimate |an (h)| ≤ dn for the n-th Taylor coefficient of h in the dual set of UST with a √ sharp constant d = M ≈ 1.2557, where M ≈ 1.5770 is the maximum value of a certain trigonometric expression. Using this, he showed that ∞ X

1 n|an | ≤ √ ⇒ f ∈ UST . M n=2 √ The bound 1/ M is sharp. Open Problem 2.4. Rønning [53] proved that UST 6⊂ S ∗ (1/2) and posed the problem of determining the largest α such that UST ⊂ S ∗ (α). Nezhmetdinov [34] showed

that UST 6⊂ S ∗ (α0 ) for some α0 ≈ 0.1483. Determine the largest α such that UST ⊂ S ∗ (α). 3. Uniformly Convex Functions

3.1. Analytic characterizations and parabolic starlike functions. Recall that a univalent function f is in the class UCV of uniformly convex functions if for every circular arc γ contained in D with center ζ ∈ D the image arc f (γ) is convex. From this definition, the following theorem is readily obtained. Theorem 3.1 ([15, Theorem 1, p. 88]). The function f belongs to UCV if and only if f ′′ (z) (3.1) Re 1 + (z − ζ) ′ ≥ 0, f (z)

z, ζ ∈ D.

Though the class C is a linear invariant family, the class UCV is not. This was proved by Goodman [15,

Theorem 5, p. 90] by using the function z . F (z) = 1 − Az

This function F ∈ UCV if and only if |A| ≤ 1/3. From the geometric definition or from Theorem 3.1, it is evident that UCV ⊂ CV. However, by taking ζ = −z

5

in Theorem 3.1, it is evident that UCV ⊂ C(1/2). In view of this inclusion and the coefficient estimate for functions

The class C of convex functions and the class S ∗ of starlike functions are connected by the Alexander result

in C(1/2), the Taylor coefficients an of f ∈ UCV satisfy |an | ≤ 1/n. Unlike the uniformly starlike functions, uni-

that f ∈ C if and only if zf ′ ∈ S ∗ . Such a question between the classes UST and UCV is in fact a question

zation, and this readily yields several important properties of functions in UCV. This one-variable characteriza-

51]) that that there is no inclusion between them:

formly convex functions admit a one-variable characteri-

tion, obtained independently by Rønning [50, Theorem 1, p. 190], and Ma and Minda [29, Theorem 2, p. 162], is the following result. Theorem 3.2. Let f ∈ A. Then f ∈ UCV if and only if ′′ zf (z) zf ′′ (z) , z ∈ D. > ′ (3.2) Re 1 + ′ f (z) f (z)

If f ∈ UCV, then equation (3.2) follows from (3.1) for a suitable choice of ζ. For the converse, the minimum principle for harmonic function is used to restrict ζ and z to |ζ| < |z| < 1. With this restriction, (3.1)

immediately follows from (3.2). To give a nice geometric interpretation of (3.2), let Ωp := {w ∈ C : Re w > |w − 1|}. The set Ωp is the interior of the parabola (Im w)2 = 2 Re w − 1 and it is therefore symmetric with respect to the real axis

and has (1/2, 0) as its vertex. Then f ∈ UCV if and only if zf ′′ (z) ∈ Ωp . 1+ ′ f (z) A class closely related to the class UCV is the class

of equality between UST and SP . It turns out (see [15, UST 6⊂ SP

functions consists of functions f ∈ A satisfying ′ ′ zf (z) zf (z) > − 1 , z ∈ D. Re f (z) f (z)

In other words, the class SP consists of function f = zF ′

where F ∈ UCV.

and

and the sector

The proof follows readily from the implication 1 1 1 ⇒ |w| < = 1 − < 1 − |w| < Re(1 + w). 2 2 2 P∞ A function f (z) = z − n=2 an z n with an ≥ 0 is called a function with negative coefficients. For functions with |w| <

negative coefficients, the above condition is also necessary for a function f to be in UCV or SP (see [11, 61]).

In terms of the coefficients, the results can be stated as follows: Theorem 3.3. Let f be a function of the form f (z) = P∞ z − n=2 an z n with an ≥ 0. Then f ∈ UCV ⇔

and f ∈ SP ⇔

∗ SP ⊂ S ∗ (1/2) ∩ S1/2 .

∞ X

n=2 ∞ X

n(2n − 1)an ≤ 1

(2n − 1)an ≤ 1.

n=2

Denote the class of all functions with negative coefficients by T . Define T UCV := T ∩ UCV, T S ∗ := T ∩ S ∗ ,

T S P := T ∩ SP , and

T C := T ∩ C.

In terms of these classes, the above result can be stated as T UCV = T C(1/2)

and

T S P = T S ∗ (1/2).

For these and other related results, see [11, 61]. Using Theorem 3.3, it can be seen [50] that f (z) = z − An z n ∈ SP ⇔ |An | ≤

{w : | arg w| < π/4}, Rønning [50] noted that

′ zf (z) 1 < ⇒ f ∈ SP . − 1 f (z) 2

(3.4)

Since the parabolic region Ωp is contained in the halfplane {w : Re w > 1/2}

SP 6⊂ UST .

3.2. Examples. To give some examples of functions in UCV and SP , note [40] that ′′ zf (z) 1 (3.3) f ′ (z) < 2 ⇒ f ∈ UCV

of parabolic starlike functions defined below.

Definition 3.1. [50] The class SP of parabolic starlike

and

and f ∈ UCV ⇔ |An | ≤

1 , 2n − 1

1 . n(2n − 1)

6

maps C onto the parabolic region

Goodman [15] showed ∞ X

n=2

∞ X 1 ⇒ f (z) = z + an z n ∈ UCV; 3 n=2

n(n − 1)|an | ≤

this easily follows from Theorem 3.3 since ∞ X

n=2

n(2n − 1)an ≤ 3

∞ X

n=2

n(n − 1)|an | ≤ 1.

The sufficient condition in (3.3) can be extended to a more general circular region. For this purpose, let a >

Ωp := {w ∈ C : Re w > |w − 1|}. Therefore the classes UCV and SP can be expressed in the form zf ′ (z) SP = f ∈ A : ≺ ϕp (z) f (z) and zf ′′ (z) UCV = f ∈ A : 1 + ′ ≺ ϕp (z) . f (z)

1/2. Then it can be shown that the minimum distance from the point w = a to points on the parabola

Rønning [50, Theorem 6, p. 195] went on to show the sharp inequality

|w − 1| = Re w

′

|f (z)| ≤ exp

is given by

Thus [59]

and

a − 1 , if 21 < a ≤ Ra = √ 2 2a − 2, if a ≥ 3 . 2

results are then proved for more general classes of functions by Ma and Minda [30]. For this purpose, let φ be

an analytic function with positive real part in D, nor-

be subordinate to the function F , written f (z) ≺ F (z), if there exists an analytic function w : D → D satisfying w(0) = 0 such that f (z) = F (w(z)). If p : D → C, p(0) = 1 and Re p(z) > 0, then 1+z . 1−z

This follows since the mapping q(z) = (1 + z)/(1 − z) maps D onto the right-half plane ΩH := {w ∈ C : Re w > 0}. In this light, the classes of starlike and convex functions can be expressed as follows: zf ′ (z) 1+z ∗ S = f ∈A: ≺ f (z) 1−z 1+z zf ′′ (z) . ≺ C = f ∈A:1+ ′ f (z) 1−z Rønning [50] and Ma and Minda [29] showed that the function ϕp : D → C defined by

8 =1+ 2 π

log

√ 2 1+ z √ 1− z

2 23 44 4 z + z2 + z3 + z + ··· 3 45 105

malized by the conditions φ(0) = 1 and φ′ (0) > 0, such that φ maps the unit disk D onto a region starlike with respect to 1 that is symmetric with respect to the real axis. They introduced the following classes: zf ′ (z) ∗ ≺ ϕ(z) S (ϕ) := f ∈ A : f (z) and

zf ′′ (z) ≺ ϕ(z) . C(ϕ) = f ∈ A : 1 + ′ f (z) These functions are called Ma-Minda starlike and convex functions respectively. For special choices of ϕ, these classes become well-known classes of starlike and convex functions. For example, for the choice ϕA,B (z) =

and

2 π2

≈ 5.502

gin contained in f (D)) and rotation (the upper bound for | arg(f ′ (z))|) estimates for functions in UCV. These

3.3. Subordination and its consequences. Let f and F be analytic functions in D. Then f is said to

ϕp (z) = 1 +

tortion (bounds for |f ′ (z)|), growth (bounds for |f (z)|), covering (the radius of the largest disk centered at ori-

′ zf (z) < Ra ⇒ f ∈ SP . − a f (z)

(3.5)

14ζ(3) π2

for f ∈ UCV, where ζ(t) denotes the Riemann zeta function. Ma and Minda [29] on the other hand obtained dis-

3 2

′′ 1 + zf (z) − a < Ra ⇒ f ∈ UCV ′ f (z)

p(z) ≺

1 + Az , 1 + Bz

−1 ≤ B < A ≤ 1,

the class S ∗ [A, B] := S ∗ (ϕA,B ) is the class of Janowski starlike functions. For the classes of Ma-Minda starlike and convex functions, the following theorem is obtained.

Theorem 3.4. [30] If f ∈ C(ϕ), then, for |z| = r,

kϕ′ (−r) ≤|f ′ (z)| ≤ kϕ′ (r), −kϕ (−r) ≤ |f (z)| ≤ kϕ (r),

7

where kϕ : D → C is defined by 1+

zkϕ′′ (z) = ϕ(z). kϕ′ (z)

Equality holds for some z 6= 0 if and only if f is a rotation

of kϕ . Also either f is a rotation of kϕ or f (D) contains the disk |w| ≤ −kϕ (−1), where −kϕ (−1) = lim (−kϕ (−r)). r→1−

Further, for |z0 | = r < 1, | arg(f ′ (z0 ))| ≤ max | arg kϕ′ (z)|. |z|=r

The proof relies on the subordination f ′ (z) ≺ kϕ′ (z) satisfied by functions f ∈ C(ϕ). Corresponding results for functions in S ∗ (ϕ) were also obtained by Ma and Minda [30]. The distortion theorem for f ∈ S ∗ (ϕ) re-

quires some additional assumptions on ϕ. Theorem 3.4

contains the corresponding results for uniformly convex functions [29] as special cases. Extension of these (and other closely related) results to functions starlike with respect to symmetric points, conjugate points, multiva-

3.4. Coefficient Problems. As noted earlier, the inclusion UCV ⊂ C(1/2) shows that each Taylor coeffi-

cient an of f ∈ UCV satisfies |an | ≤ 1/n. These bounds can be improved. Since the classes UCV and SP are con-

nected by the Alexander relation that f ∈ UCV if and only if zf ′ ∈ SP , it suffices to give the coefficient estimate for functions in SP .

Theorem 3.6. [50, Theorem 5, p. 194] Let f ∈ SP and P∞ f (z) = z + n=2 an z n . Then (3.6) n c Y c |a2 | ≤ c, and |an | ≤ , 1+ n−1 k−2 k=3

where c = 8/π 2 .

Let p(z) = zf ′ (z)/f (z) = 1 + c1 z + c2 z 2 + · · · , and

p(z) ≺ ϕp (z) where ϕp is given by (3.5). Rogosinski’s

theorem states that |ck | ≤ c for any function p(z) = 1 + c1 z + c2 z 2 + · · · subordinate to the convex univalent

function P (z) = 1 + cz + · · · . The coefficients of f and the coefficients of p are related by

lent starlike functions, and meromorphic functions were investigated in [43, 6, 5]. Let hϕ : D → C be defined by zh′ϕ(z) hϕ (z)

= ϕ(z).

f (z) hϕ (z) ≺ . z z In the case when ϕ is a convex univalent function, this result is a special case of the following general result: f ∈ S ∗ (ϕ) ⇒

Theorem 3.5 (Ruscheweyh [55, Theorem 1, p. 275]). Let φ be a convex function defined in D with φ(0) = 1. Define F by F (z) = z exp

z 0

n−1 X

cn−k ak .

k=1

This together with Rogosinksi’s theorem yield the desired

Ma and Minda [30] proved that

Z

(n − 1)an =

φ(x) − 1 dx . x

The function f belongs to S ∗ (φ) if and only if for all |s| ≤ 1 and |t| ≤ 1, sF (tz) sf (tz) ≺ . tf (sz) tF (sz)

coefficient bounds. Whenever ϕ is a convex univalent function, the bounds for |an | for f ∈ S ∗ (ϕ) is also given

by (3.6) where c := ϕ′ (0). The estimates given by (3.6) are not sharp in general. However, in the case, ϕ(z) = (1+z)/(1−z), the inequalities in (3.6) give sharp bounds for the coefficients of starlike functions.

The sharp coefficient estimates for functions in UCV or SP is still an open problem. However, the sharp esti-

mates of |an | for f ∈ UCV were obtained by Ma and

Minda [29, 31]. They [29, Theorem 5, p. 172] also proved the sharp order of growth |an | = O(1/n2 ) for

f ∈ UCV. The same order of growth holds for f ∈ C(ϕ) if ϕ belongs to the Hardy class of analytic functions H2

(see [30]). They [31] also found the sharp upper bound for the Fekete-Szeg˝ o functional |µa22 − a3 | in the class

UCV for all real µ. For the inverse function f −1 (w) = w +

∞ X

dn wn ,

n=2

Open Problem 3.1. Determine the sharp bound of (n)

they [31] obtained the sharp inequality

∗

|f (z)| for f ∈ C(ϕ) and f ∈ S (ϕ). For f ∈ C(ϕ), the bounds for the cases n = 0, 1 are given by Theorem 3.4. Similar bounds for f ∈ S ∗ (ϕ) are also known with some restrictions on ϕ.

|dn | ≤

8 , (n − 1)nπ 2

n = 2, 3, 4.

More generally, the coefficient problem for f ∈ C(ϕ) is also open.

Estimates for the first two coefficients as

8

well as for the Fekete-Szeg˝ o functional for functions in C(ϕ) were obtained in [30]. For several related coeffi-

3.5. Convolution. Recall that the convolution of two analytic functions

cient problems, see [7].

f (z) = z + Theorem 3.7. Let φ(z) = 1 + B1 z + B2 z 2 + · · · . If 2

3

f (z) = z + a2 z + a3 z + · · · ∈ C(ϕ), then 1 2 2 6 (B2 − (3/2)µB1 + B1 ) if 3B12 µ ≤ 2(B2 + B12 − B1 ) B1 if 2(B2 + B12 − B1 ) ≤ 3B12 µ 6 2 |a3 − µa2 | ≤ ≤ 2(B2 + B12 + B1 ) 1 2 2 6 (−B2 + (3/2)µB1 − B1 ) if 2(B + B 2 + B ) ≤ 3B 2 µ 2 1 1 1 The result is sharp.

To see an outline of the proof, first express the coefficient of f in terms of the coefficients ck for functions with positive real part. For f ∈ C(ϕ), let p : D → C be defined by

p(z) :=

zf ′ (z) = 1 + b1 z + b2 z 2 + · · · f (z)

so that 2a2 = b1 and 6a3 = b2 + b21 . Since φ is univalent and p(z) ≺ φ(z), the function p1 (z) =

1 + φ−1 (p(z)) = 1 + c1 z + c2 z 2 + · · · 1 + φ−1 (p(z))

is analytic and has positive real part in D. Also p1 (z) − 1 (3.7) p(z) = φ p1 (z) + 1 and from this equation (3.7), it follows that 1 1 1 1 B1 c1 and b2 = B1 (c2 − c21 ) + B2 c21 . 2 2 2 4

b1 = Therefore

a3 − µa22 =

(3.8) where v :=

1 2B1

B1 c2 − vc21 , 12

∞ X

an z n ,

and

g(z) = z +

n=2

∞ X

bn z n

n=2

is the analytic function defined by (f ∗ g)(z) := z +

∞ X

an b n z n .

n=2

The convolution of two functions in A is again in A. Since the nth coefficient of normalized univalent function is bounded by n, the convolution of the Koebe function k(z) = z/(1 − z)2 with itself is not univalent. Thus, the

convolution of two univalent (or starlike) functions need not be univalent. P´ olya and Schoenberg [38] conjectured that the class of convex functions C is preserved under

convolution with convex functions:

f, g ∈ C ⇒ f ∗ g ∈ C. In 1973, Ruscheweyh and Sheil-Small [58] (see also [57]) proved the Polya-Schoenberg conjecture. In fact, they also proved that the classes of starlike functions and close-to-convex functions are closed under convolution with convex functions. The proof of these facts follow from the following result which is also used below to show that the classes UCV and SP are closed under convolution with starlike functions of order 1/2. Theorem 3.8. [58, Theorem 2.4, p. 54] Let α ≤ 1, f ∈ Rα and g ∈ S ∗ (α). Then, for any analytic function H ∈ H(D),

f ∗ Hg (D) ⊂ co(H(D)), f ∗g where co(H(D)) denote the closed convex hull of H(U ). Theorem 3.9. [49, Theorem 3.6, p. 131] Let ϕ be a

3 B1 − B12 − B2 + µB12 . 2

convex function with Re ϕ(z) ≥ α, α < 1. If f ∈ Rα and g ∈ S ∗ (ϕ), then f ∗ g ∈ S ∗ (ϕ).

The theorem then follows by an application of the corresponding coefficient results for function with positive real

The proof of this theorem follows readily from Theorem 3.8 by putting H(z) = zg ′ (z)/g(z). In view of

part. Notice that this method is difficult to apply to get bounds for |an | for large n, as an can only be expressed

the fact that f ∈ C(ϕ) if and only if zf ′ ∈ S ∗ (ϕ), an immediate consequence of the above theorem is the cor-

as a non-linear function of the coefficients ck .

Open Problem 3.2. Determine the sharp bound for the Taylor coefficients |an | (n ≥ 5) for f ∈ C(ϕ) and f ∈ ∗

S (ϕ). The same problem for the other classes defined by subordination is still open.

responding result for C(ϕ): if f ∈ Rα and g ∈ C(ϕ), then f ∗ g ∈ C(ϕ) for any convex function ϕ with Re ϕ(z) ≥ α.

In particular, the classes UCV and SP are closed under

convolution with starlike functions of order 1/2. Similar results for several other related classes of functions can be found in [3, 4, 42] or references therein.

9

Goodman remarked that the class UCV is preserved under the transformation e−iα f (eiα z) and no other transformation seems to be available. However, since UCV is closed under convolution with starlike functions of order 1/2 and in particular with convex functions, the following result is obtained.

Γ1 (f (z)) = zf ′ (z), 1 [f (z) + zf ′ (z)] 2 Z k + 1 z k−1 ζ f (ζ)dζ, Re k > 0 Γ3 (f (z)) = zk 0 Z z f (ζ) − f (ηζ) Γ4 (f (z)) = dζ, |η| ≤ 1, η 6= 1. ζ − ηζ 0 Γ2 (f (z)) =

Then Γi (f ) ∈ UCV in |z| < ri whenever f ∈ UCV, where √ 17 − 3 1 ≈ .56155, r3 = r4 = 1. r1 = , r2 = 3 2 3.6. Gaussian Hypergeometric functions. For complex numbers a, b, c ∈ C with c 6= 0, −1, −2, . . . , the

Gaussian hypergeometric function F (a, b; c; z) is defined by the power series ∞ X (a)n (b)n z n . F (a, b, c; z) := (c)n n! n=0

Theorem 3.11. [27, Theorem 4, p. 771] Let a, b ∈ C −

{0} and c > |a| + |b| + 1. If

Γ(c − |a| − |b|)Γ(c) × Γ(c − |a|)Γ(c − |b|) ! 2|ab| 1+ − 1 ≤ 1, c − |a| − |b| − 1

and f ∈ Rη (β), then zF (a, b; c; z) ∗ f (z) ∈ UCV. An extension of these results to other related classes can be found, for example, in [19, 25]. 3.7. Integral transform. The classes UCV and SP

are closed under several integral operators.

Theorem 3.12. [59, Theorem 1, p. 320] Let fi ∈ UCV Pn 1 αi ≤

and αi ’s be real numbers such that αi ≥ 0, and 1. Then the function Z zY n [fi′ (ζ)]αi dζ g(z) = 0 i=1

belongs to UCV.

As an immediate consequence of this theorem, the

Here (a)0 := 1 for a 6= 0 and if n is a positive integer, then (a)n := a(a + 1)(a + 2) · · · (a + n − 1). For β < 1

and η ∈ R, define the class Rη (β) by Rη (β) = f ∈ A | Re(eiη f ′ (z) − β) > 0 for

z∈D .

For the Gaussian hypergeometric function F (a, b, c; z), Kim and Ponnusamy [27] found conditions which would imply that zF (a, b; c; z) belongs to UCV or Rη (β). Further they derived conditions under which f ∈ Rη (β) im-

plies

zF (a, b; c; z) ∗ f (z) ∈ UCV. In fact, by making use of the Gauss summation theorem and Theorem 3.3, they obtained the following sufficient condition for zF (a, b; c; z) ∈ UCV. Theorem 3.10. [27, Theorem 1, p. 768] Let a, b ∈ C − {0} and c > |a| + |b| + 2. If Γ(c − |a| − |b|)Γ(c) × Γ(c − |a|)Γ(c − |b|) 2(|a|)2 (|b|)2 5|ab| + (c − 2 − |a| − |b|)2 c − |a| − |b| − 1 then zF (a, b; c; z) ∈ UCV.

following result provides a mapping of Rη (β) into UCV.

2(1 − β) cos η

Corollary 3.1. [40] Let

1+

They also obtained a weaker condition on the parameters so that the function zF (a, a; c; z) ∈ UCV. The

function g defined by Z g(z) =

0

(0 ≤ αi < 1,

n X 1

n zY 1

(1 − Ai ζ)−2αi dζ

αi ≤ 1, |Ai | ≤

1 , i = 1, 2, . . . , n) 3

belongs to UCV. The first implication in (3.3) yields the

following result.

Theorem 3.13. [59, Theorem 2, p. 320] If f ∈ A satisfies ′ 1 zf (z) f (z) − 1 < 4 , then 2 Z z f (ζ) g(z) = dζ ζ 0 belongs to UCV. 3.8. k-Uniformly convex function. Let k ≥ 0.

A function f ∈ S is called k-uniformly convex in D if the image of every circular arc γ contained in the unit disk

D, with center ζ, |ζ| ≤ k, is convex. For any fixed k ≥ 0,

≤ 2,

the class of all k-uniformly convex functions is denoted by k − UCV. The class k − UCV was introduced and

investigated by Kanas and Wisinowska [22]. As in the

10

case of uniformly convex functions, the following theorem holds. Theorem 3.14 ([22]). Let f ∈ S. Then the following

are equivalent:

arc Γz with center at ζ lying in D is convex α-spirallike. The class of all uniformly convex α-spiral functions is de-

(2) the inequality f ′′ (z) ≥0 Re 1 + (z − ζ) ′ f (z) holds for all z ∈ D and for all |ζ| ≤ k, (3) the inequality ′′ zf (z) zf ′′ (z) Re 1 + ′ > k ′ f (z) f (z) holds for all z ∈ D.

Interestingly, the class of k-uniformly convex functions unifies the class of convex functions (k = 0) and the class of uniformly convex functions (k = 1). Let Ωk = {w : Re w > k|w − 1|}. Then the region Ωk is elliptic for k > 1, parabolic for k = 1, and hyperbolic for 0 < k < 1. The region Ω0 is the right-half plane. Several properties of uniformly convex functions extend to k − UCV functions; these properties are treated in [22, 23, 17, 19, 20, 21]. 3.9. Uniformly spirallike functions. Let Γw be the image of an arc Γz : z = z(t), (a ≤ t ≤ b) under the function f (z) and let w0 be a point not on Γw . Recall that the arc Γw is starlike with respect to w0 if arg(w − w0 ) is a nondecreasing function of t. This condition is equivalent to f ′ (z)z ′ (t) ≥0 f (z) − w0

noted by UCSP(α). An analytic description of UCSP(α) analogous to the class UCV is the following: Theorem 3.16. [46] Let f ∈ A. The the following are equivalent. (1) f ∈ UCSP(α), (2) f satisfies the inequality (z − ζ)f ′′ (z) ≥ 0, Re e−iα 1 + f ′ (z)

z 6= ζ,

(3) f satisfies the inequality ′′ zf (z) zf ′′ (z) , ≥ ′ Re e−iα 1 + ′ f (z) f (z)

z, ζ ∈ D,

z ∈ D.

For f ∈ A, define the function s by iα

f ′ (z) = (s′ (z))e

cos α

.

Then f ∈ UCSP(α) if and only if s ∈ UCV. In view

of this connection with UCV, properties of functions in UCSP can be obtained from the corresponding properties of UCV. The classes of uniformly spirallike and uniformly convex spirallike functions were introduced by

Ravichandran et al. [46], and for a generalization of the class, see [63]. 3.10. Radius problems. The determination of the radius of starlikeness or convexity typically requires an estimate for the real part of the quantities

(a ≤ t ≤ b).

The arc Γw is α-spirallike with respect to w0 if arg

lies between α and α + π. The function f is a uniformly convex α-spiral function if the image of every circular

(1) f ∈ k − UCV,

Im

The arc Γw is convex α-spirallike if ′′ z (t) z ′ (t)f ′′ (z) arg + z ′ (t) f ′ (z)

z ′ (t)f ′ (z) f (z) − w0

QST :=

zf ′ (z) f (z)

and QCV := 1 +

zf ′′ (z) . f ′ (z)

This method of estimating the real part of QST or QCV will not work for the radius problems associated with

lies between α and α + π [14]. The function f is uniformly α-spirallike if the image of every circular arc Γz

uniformly convex functions, parabolic starlike functions, strongly starlike functions and several other subclasses

with center at ζ lying in D is α-spirallike with respect to f (ζ). The class of all uniformly α-spirallike functions is

of starlike/convex functions. In these cases, one need to know the region of values of QST or QCV . This idea was

denoted by USP(α). Here is an analytic description of USP(α) analogous to the class UST . Theorem 3.15. [46] Let |α| <

π 2.

A function f ∈ A

belongs to USP(α) if and only if (z − ζ)f ′ (z) ≥ 0, z 6= ζ, Re e−iα f (z) − f (ζ)

z, ζ ∈ D.

first used by Rønning for computing the sharp radius of parabolic starlikeness for univalent functions. Theorem 3.17. The SP -radius of the class S of univa-

lent functions is 0.33217 and the SP -radius of the class S ∗ of starlike functions is 1/3 ≈ 0.3333 [50, Corollary

3, Theorem 4, p. 192]. The SP -radius of the class C of

11

√ convex functions is 1/ 2 ≈ 0.7071 [51, Theorem 3.1 9b, p. 236].

function f (z) = z + (

Nδ (f ) := The SP -radii for the following classes of functions

were determined by Shanmugam and Ravichandran [59]:

starlike of order α. (2) the class of functions f (z) = z + an z n + · · ·

satisfying the condition Re(f (z)/z) > 0. (3) the class of functions f ∈ A satisfying f (z) g(z) − 1 < 1

f (z) = z + an+1 z n+1 + an+2 z n+2 + · · · satisfying

1 + Az zf ′ (z) ≺ . f (z) 1 + Bz For the special case A = 1 − 2α, B = −1, the class is denoted by Sn∗ (α). Ravichandran, Rønning and ShanSn∗ (β)-radius

mugam [47] investigated and SP -radius for the class Sn∗ [A, B]. They also investigated the radii of Sn∗ (0).

convexity and uniform convexity in Additionally they studied the radius problems for functions whose

an z n ∈ A to be the set

∞ X

k

bk z and

k=2

∞ X

k=2

)

k|ak −bk | ≤ δ .

Ruscheweyh [56] proved among other results that N1/4 (f ) ⊂ S ∗ for f ∈ C. For a more general notion of T -δ-neighbour-

hood of an analytic function, see Sheil-Small and Silvia [60]. Padmanabhan [35] investigated the neighbourhood problem for the class UCV. Since the class UCV is closed under convolution with starlike functions of order 1/2, it follows that the function (f (z) + ǫz)/(1 + ǫ) ∈ SP for |ǫ| < 1/4. Using f ∈ SP ⇔

for some function g starlike of order α.

Rønning [53, Theorem 4, p. 321] showed that the radius of uniform convexity of the classes S and S ∗ is √ (4 − 13)/3 ≈ 0.1314. Let Sn∗ [A, B] consists of functions

n=2

g : g(z) = z+

(1) the class of close-to-starlike functions of order α; these are functions f ∈ A satisfying the condition Re(f (z)/g(z)) > 0 for some function g

P∞

1 (f ∗ h)(z) 6= 0, z

t ∈ R, z ∈ D,

where h(z) :=

2 1 − 2it − t2

z − (1 − z)2

t2 + 1 + it 2

z 1−z

,

Padmanabhan proved that Nδ (f ) ⊂ SP whenever f (z) + ǫz ∈ SP 1+ǫ for |ǫ| < δ < 1. These two assertions together show that N1/8 (f ) ⊂ SP for f ∈ UCV. For some related results, see [17]. References [1] R. Aghalary and S. R. Kulkarni, Certain properties of parabolic starlike and convex functions of order ρ, Bull. Malays. Math. Sci. Soc. (2) 26 (2003), no. 2, 153–162.

derivatives belong to the Kaplan classes K(α, β); their results, in special cases, yield radius results for vari-

[2] R. M. Ali, Starlikeness associated with parabolic regions, Int.

ous classes of close-to-convex functions and functions of

[3] R. M. Ali, A. O. Badghaish and V. Ravichandran, Multivalent

bounded boundary rotation. For 0 ≤ α ≤ β, the Kaplan classes K(α, β) can be defined as follows. A func′

tion f ∈ K(α, β) if and only if there is a function g ∈ S ∗ ((2 + α − β)/2) and a real number t ∈ R such that ′ arg eit zf (z) ≤ απ . g(z) 2

For the radius of uniform convexity of a closely related

class, see [48] wherein they investigated S ∗ (β)-radius and SP -radius of certain integral transforms and Bloch

functions. Related radius results can also be found in [45].

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cheweyh [56] defined the δ-neighbourhood Nδ (f ) of a

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School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia E-mail address: [email protected]

Department of Mathematics, University of Delhi, Delhi–110 007, India E-mail address: [email protected]

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