Some properties for certain classes of univalent ... - ScienceDirect

Acta Mathematica Scientia 2017,37B(1):69–78 http://actams.wipm.ac.cn

SOME PROPERTIES FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS DEFINED BY DIFFERENTIAL INEQUALITIES∗

$“f)

Zhigang PENG (

¨2)

Gangzhen ZHONG (

Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China E-mail : [email protected]; [email protected] Abstract Let A be the space of functions analytic in the unit disk D = {z : |z| < 1}. Let U denote the set of all functions f ∈ A satisfying the conditions f (0) = f ′ (0) − 1 = 0 and z 2 ′ ) − 1 < 1 (|z| < 1). f (z)( f (z) Also, let Ω denote the set of all functions f ∈ A satisfying the conditions f (0) = f ′ (0) − 1 = 0 and 1 |zf ′ (z) − f (z)| < (|z| < 1). 2 In this article, we discuss the properties of U and Ω. Key words

univalent function; Starlike function; Hadamard product; extreme point; support point

2010 MR Subject Classification

1

30C45; 30C75; 30C80

Introduction

Let A be the space of functions analytic in the unit disk D = {z : |z| < 1}. Endowed with the topology of uniform convergence on compact subsets of the unit disk, A is a locally convex topological vector space. Let B be the subset of A which consists of φ ∈ A with |φ(z)| ≤ 1 (|z| < 1). Suppose that F is a compact subset of A. A function f is called a support point of F if f ∈ F and there is a continuous linear functional J on A such that ReJ is non-constant on F and ReJ(f ) = max{ReJ(g) : g ∈ F }. The set of all support points of F is denoted by suppF . An element f ∈ F is called an extreme point of F if it is not a proper convex combination of any two distinct points in F . The set of extreme points of F is denoted by EF . In recent decades the extreme points and support points of many classes of analytic functions were studied (see [1–12]). ∗ Received November 6, 2015; revised March 18, 2016. Mathematics in Hubei Province, China.

Supported by the Key Laboratory of Applied

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Let S be the subset of A consisting of functions f that are univalent in D and satisfy f (0) = f ′ (0) − 1 = 0. A function f ∈ S is called starlike if f (D) is starlike with respect to the origin. The class of all starlike functions is denoted by S ∗ . A function f ∈ S ∗ if and only if Re

zf ′ (z) >0 f (z)

(|z| < 1).

A function f ∈ S is called convex if f (D) is a convex set. The class of all convex functions is denoted by K. A function f ∈ K if and only if   zf ′′ (z) Re 1 + ′ > 0 (|z| < 1). f (z) In recent years, many scholars were still interested in studying the properties of various subclasses of univalent functions (see [13–20]). Let U denote the set of all functions f ∈ A satisfying the conditions f (0) = f ′ (0) − 1 = 0 and ′ f (z)( z )2 − 1 < 1 (z ∈ D). (1.1) f (z) Let h(z) =

z f (z)

− 1. It is easy to show that f ∈ U if and only if zh′ (z) − h(z) = z 2 φ(z),

(1.2)

where φ ∈ B, that is, φ is analytic in D and |φ(z)| ≤ 1 for all z ∈ D. It is well known that the functions in U are univalent [15]. Up to now, the class U were studied in detail for many years (see [16–20]). In the second part of this paper, we further study the class U, and meanwhile, we want to study the univalence of the function h(z) which is a solution of the differential equation (1.2). For this purpose, we define a new class Ω which consists of functions f ∈ A satisfying the conditions f (0) = f ′ (0) − 1 = 0 and 1 (z ∈ D). 2 It is clear that inequality (1.3) is equivalent to the equation |zf ′ (z) − f (z)|
0 f (z)

for |z| < 1 and so, f ∈ S ∗ .



Remark 3.2 According to Theorem 3.1, if f (z) ∈ A, f (0) = f ′ (0) − 1 = 0 and |zf ′ (z) − f (z)| < λ, then f ∈ S ∗ ⊂ S when λ ≤ 12 . But f may be not univalent when λ > 12 . One can find it by studying the function f (z) = z + 21 λz 2 . In order to prove Theorem 3.4, we need the following lemma which is due to Dieudonn´e (see [21], pp.198–199). Lemma 3.3 Let z0 and w0 be given points in D, with z0 6= 0. Then for all functions f analytic and satisfying |f (z)| < 1 in D, with f (0) = 0 and f (z0 ) = w0 , the region of values of f ′ (z0 ) is the closed disk 2 2 w − w0 ≤ |z0 | − |w0 | . (3.8) z0 |z0 |(1 − |z0 |2 ) Theorem 3.4 The radius of convexity for Ω is 21 .

Proof

Suppose that f ∈ Ω. Then, by (3.4), Z z z f (z) = z + ϕ(ζ)dζ, 2 0

where ϕ ∈ A and |ϕ| ≤ 1. A calculation shows that f ′′ (z) =

1 1 ϕ + [zϕ(z)]′ . 2 2

Using Lemma 3.3 for zϕ(z), we have   1 1 |z|2 − |zϕ(z)|2 |f (z)| ≤ |ϕ(z)| + |ϕ(z)| + . 2 2 |z|(1 − |z|2 ) ′′

Now, let |z| = r, |ϕ(z)| = R. Then it follows from (3.2) and (3.9) that ′′ zf (z) r2 + 2r(1 − r2 )R − r2 R2 . f ′ (z) ≤ 2(1 − r)(1 − r2 )

(3.9)

(3.10)

Let

Ψ(R) = r2 + 2r(1 − r2 )R − r2 R2 . Then Ψ(R) attains its maximum at R =

1−r 2 r .

But R =

1−r 2 r

Ψ(R) ≤ Ψ(1) = 2r(1 − r2 ) if r ≤

√

5−1 2 .

Especially,

Therefore, for r ≤

√ 5−1 2 ,

we have ′′ zf (z) r f ′ (z) ≤ 1 − r . ′′ zf (z) f ′ (z) < 1

< 1 if and only if r >

√ 5−1 2 .

So

No.1

Z.G. Peng & G.Z. Zhong: SOME PROPERTIES FOR CERTAIN CLASSES

if r < 12 , and so,

73

  zf ′′ (z) Re 1 + ′ >0 f (z)

for r < 21 . This implies that f (z) is convex in |z| < 12 . On the other hand, f (z) = z + 12 z 2 ∈ Ω and the following inequality    1 + 2z  zf ′′ (z) Re 1 + ′ = Re >0 f (z) 1+z holds only in the disk |z| < 21 . So, we conclude that the radius of convexity for Ω is 12 .



Definition 3.5 The convolution, or Hadamard product of two functions f (z) =

∞ X

an z n (|z| < 1)

∞ X

bn z n (|z| < 1)

n=0

and g(z) =

n=0

is the function h = f ∗ g with h(z) =

∞ X

an bn z n (|z| < 1).

n=0

Lemma 3.6 If φ, ϕ ∈ B then φ ∗ ϕ ∈ B. Proof

Since φ, ϕ ∈ B, we have |φ(z)| ≤ 1, |ϕ(z)| ≤ 1. Let h = φ ∗ ϕ. Then Z 2π 1 2 iθ h(r e ) = φ(rei(θ−t) )ϕ(reit )dt (r < 1). 2π 0

Thus |h(r2 eiθ )| ≤

1 2π

Z

2π

|φ(rei(θ−t) )| · |ϕ(reit )|dt ≤ 1 (r < 1).

0

That is, φ ∗ ϕ ∈ B.



Theorem 3.7 If f, g ∈ Ω then f ∗ g ∈ Ω. Proof

Since f, g ∈ Ω, there exist φ, ϕ ∈ B such that 1 f (z) = z + 2

Z

1

1 2

Z

1

g(z) = z +

z 2 φ(zt)dt,

(3.11)

z 2 ϕ(zt)dt.

(3.12)

0

0

So   Z 1 1 2 (f ∗ g)(z) = z + z φ(zt)dt ∗ g(z) 2 0 Z 1 1 2 = z ∗ g(z) + [z φ(zt)] ∗ g(z)dt 2 0    Z Z 1 1 2 1 1 2 =z+ [z φ(zt)] ∗ z + z ϕ(zλ)dλ dt 2 0 2 0  Z Z 1 1 1 =z+ [z 2 φ(zt)] ∗ [z 2 ϕ(zλ)]dλ dt 4 0 0

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1 =z+ 4

Z

1

1 =z+ 2

Z

1

where

0

Z

1

0

Vol.37 Ser.B

 z [φ(zt) ∗ ϕ(zλ)]dλ dt 2

z 2 ω(zt)dt,

0

1 ω(z) = 2

Z

1

φ(z) ∗ ϕ(zλ)dλ.

0

By Lemma 3.6, we have |ω(z)| ≤

1 2

Z

1

|φ(z) ∗ ϕ(zλ)|dλ ≤

0

1 < 1. 2

Thus, f ∗ g ∈ Ω.



Theorem 3.8 Ω is a closed convex subset of A. Proof It is a clear that Ω is convex. We next show that Ω is closed. Suppose that fn ∈ Ω and {fn } converges to f uniformly on compact subsets of unit disk. Then 1 |zfn′ (z) − fn (z)| < , n = 1, 2, 3, · · · . 2 Letting n → ∞, we get 1 |zf ′ (z) − f (z)| ≤ . 2 If |zf ′ (z) − f (z)| = 12 for some z ∈ D then zf ′ (z) − f (z) ≡ 21 λ, where |λ| = 1. But it is not the case, since zf ′ (z) − f (z) is zero when z = 0. So |zf ′ (z) − f (z)|