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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(3): 513-522

Published online September 30, 2021

### On Coefficients of a Certain Subclass of Starlike and Bistarlike Functions

Hesam Mahzoon, Janusz SokóŁ*

Department of Mathematics, Islamic Azad University, West Tehran Branch, Tehran, Iran
e-mail : mahzoon_hesam@yahoo.com

College of Natural Sciences, University of Rzeszow, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
e-mail : jsokol@ur.edu.pl

Received: June 6, 2019; Revised: June 22, 2020; Accepted: July 2, 2020

### Abstract

In this paper we investigate a subclass M(α) of the class of starlike functions in the unit disk |z| < 1. M(α), π/2 ≤ α < π, is the set of all analytic functions f in the unit disk |z| < 1 with the normalization f(0) = f′(0) − 1 = 0 that satisfy the condition 1+απ2sinα<Re{zf'(z)f(z)}<1+α2sinα(zΔ) The class M(α) was introduced by Kargar et al. [Complex Anal. Oper. Theory 11: 1639–1649, 2017]. In this paper some basic geometric properties of the class M(α) are investigated. Among others things, coefficients estimates and bound are given for the Fekete-Szegö functional associated with the k–th root transform [f(zk)]1/k. Also a certain subclass of bi–starlike functions is introduced and the bounds for the initial coe?cients are obtained.

Keywords: analytic functions, starlike and bistarlike functions, subordination, Fekete-Szegö, inequality.

### 1. Introduction

Let A be the class of functions f of the form

f(z)=z+ n=2anzn,

which are analytic and normalized by f(0)=f(0)1=0 in the open unit disk Δ={z:|z|<1}. The subclass of A of all univalent functions f in Δ is denoted by S. We denote by P the well-known class of analytic functions p with p(0) = 1 and Re(p(z))>0, z∈Δ. We also denote by B the class of analytic functions w(z) in Δ with w(0)=0 and |w(z)|<1, zΔ. If f and g are two functions in A, then we say that f is subordinate to g, written f(z)g(z), if there exists a function wB such that f(z)=g(w(z)) for all z∈Δ. As a special case, if the function g is univalent in Δ, then we have the following equivalence:

f(z)g(z)(f(0)=g(0)andf(Δ)g(Δ)).

A function fS is starlike (with respect to 0) if twf(Δ) whenever wf(Δ) and t[0,1]. The class of starlike functions is denoted by S*. We say that fS*(γ) (0γ<1) if and only if

Rezf(z)f(z)>γ(zΔ).

The equality S*(0)=S* is well known. Recently Kargar et al. (see [4]) introduced a certain subclass of starlike functions as follows.

Definition 1.1. Let π/2α<π. Then the function fA belongs to the class M(α) if f satisfies

1+απ2sinα<Rezf(z)f(z)<1+α2sinα  (zΔ).

Consider the function ϕ as follows

ϕ(α):=1+απ2sinα(π/2α<π).

It is clear that ϕ(π/2)=1π/40.2146 and

limαπϕ(α)=12.

Thus, the class M(α) is a subclass of the class fS*(ϕ(π/2)) of starlike functions of order ϕ(π/2)=1π/4.

By the subordination principle we have the following lemma.

Lemma 1.2. [4] Let f(z)A and π/2α<π. Then fM(α) if and only if

zf(z)f(z)1Bα(z)  (zΔ),

where

Bα(z):=12isinαlog1+zeiα1+zeiα  (zΔ).

The function Bα(z) is convex univalent in Δ and maps Δ onto

Ωα:=w:απ2sinα<Re(w)<α2sinα,

in other words, the image of Δ is a vertical strip when π/2α<π. For other α, Bα(z) is convex univalent in Δ and maps Δ onto the convex hull of three points (one of which may be that point at infinity) on the boundary of Ωα. Therefore, in other cases, we obtain a trapezium, or a triangle, see [3]. Also, we have that

Bα(z)= n=1Anzn  (zΔ),

where

An=(1)(n1)einαeinα2insinα  (n=1,2,).

The following lemma will be useful.

Lemma 1.3. (see [9]) Let q(z)= n=1Qnzn be analytic and univalent in Δ, and suppose that q(z) maps Δ onto a convex domain. If p(z)= n=1Pnzn is analytic in Δ and satisfies the following subordination

p(z)q(z)  (zΔ),

then

|Pn||Q1|  n1.

This paper is organized as follows. In Section 2 we study the class M(α). We consider the coefficient estimates and Fekete-Szegö inequality. Also, in Section 3 we introduce a certain subclass Mσ(α) of bi--univalent functions and we estimate the initial coefficients of functions belonging to Mσ(α).

### 2. Coefficient Estimates

Theorem 2.1. ([10]) Let π/2α<π. If a function fA of the form (1.1) belongs to the class M(α), then

|an|1  (n=2,3,4,).

Here, we considerthe problem of finding sharp upper bounds for the Fekete-Szegö coefficient functional associated with the k-th root transform for functions in the class M(α). For a univalent function f(z) of the form (1.1), the k-th root transform is defined by

F(z)=[f(zk)]1/k=z+ n=1bkn+1zkn+1  (zΔ).

In order to prove next result, we need the following lemma due to Keogh and Merkes [5].

Lemma 2.2. [5] Let the function g(z) given by

g(z)=1+c1z+c2z2+,

be in the class P. Then, for any complex number µ

|c2μc12|2max{1,|2μ1|}.

The result is sharp.

Theorem 2.3. Let π/2α<π. Suppose also that fM(α) and let F be the k-th root transform of f defined by (2.2). Then, for any complex number µ,

b2k+1μbk+1212kmax1,2μk1+kcosα2k.

The result is sharp.

Proof. Since fM(α), from Lemma 1.2 and by definition of subordination, there exists a function wB such that

zf(z)/f(z)=1+Bα(w(z)).

We define

p(z):=1+w(z)1w(z)=1+p1z+p2z2+,

and note that pP. Relationships (1.6) and (2.5) give us

1+Bα(w(z))=1+12A1p1z+14A2p12+12A1 p2 12p12z2+,

where A1=1 and A2=cosα. If we equate the coefficients of z and z2 on both sides of (2.4), then we get

a2=12p1,

and

a3=18(1cosα)p12+14p212p12.

Foreach f given by (1.1) and with a simple calculation we have

F(z)=[f(z1/k)]1/k=z+1ka2zk+1+1ka312k1k2a22z2k+1+.

Moreover by (2.2) and (2.9), we obtain

bk+1=1ka2andb2k+1=1ka312k1k2a22.

By inserting (2.7) and (2.8) into (2.10), we get

bk+1=p12k,

and

b2k+1=18k1cosαk1kp12+14kp212p12.

Therefore,

b2k+1μbk+12=14kp22μ+k1+kcosα2kp12.

Applying Lemma 2.2 in (2.11) with

μ=2μ+k1+kcosα2k,

gives the inequality (2.3). For the sharpness it is sufficient to consider the k-th root transforms of the function

f(z)=zexp0z Bα(w(t))tdt.

It is clear that fM(α). If we take in (2.12) w(z)=z, then from (2.5) we obtain p1=p2=2 hence from (2.11) we get

b2k+1μbk+12=12k2μk1+kcosα2k.

If we take in (2.12) w(z)=z2, then from (2.5) we obtain p1=0 while p2=2 hence from (2.11) we get for this case

b2k+1μbk+12=12k.

It shows the sharpness of (2.3) and ends the proof.

The problem of finding sharp upper bound for the coefficient functional |a3μa22| for different subclasses of the class A is known as the Fekete-Szegö problem. Putting k=1 in the Theorem (2.3) gives us:

Corollary 2.4. Let α[π/2,π). Suppose also that fM(α). Then, for any complex number µ,

a3μa2212max1,2μ2+cosα2.

The result is sharp.

Putting α=π/2, in the Corollary 2.4, we get:

Corollary 2.5. Assume that the function f given by (1.1) satisfies in the following two-sided inequality:

1π4<Rezf(z)f(z)<1+π4  zΔ,

then

a3μa2212max1,μ1  (μ).

If we take απ in the Corollary 2.4, then we have:

Corollary 2.6. Assume that the function f given by (1.1) satisfies in the following inequality:

Rezf(z)f(z)>1π4  zΔ,

then

a3μa2212max1,(2μ3)/2  (μ).

Corollary 2.7. Letthe function f, given by (1.1), be in the class M(α). Also let the function f1(w)=w+ n=2bnwn be the inverse of f. Then

|b2|1,

and

|b3|12|6cosα|  π/2α<π.

We remark that every function fS has an inverse f1, defined by f1(f(z))=z (zΔ) and

f(f1(w))=w  (|w|<r0;r01/4),

where

f1(w)=wa2w2+(2a22a3)w3(5a235a2a3+a4)w4+.

Proof Comparing (2.18) with f1(w)=w+ n=2bnwn, gives us

b2=a2andb3=2a22a3.

Applying Theorem 2.1 we get

|b2|=|a2|1.

The second inequality (2.17) follows by taking µ=-2 in the Corollary 2.4.

### 3. Bi–Univalent Functions

First, we recall that a function fA is said to be bi-univalent in Δ if f univalent in Δ and f-1 has an univalent extension from |w|<r0<1 to Δ. We denote by σ the class of bi-univalent functions in the unit disk Δ.

In 1967 Lewin [6] introduced the class σ of bi-univalent functions. He obtained the bound for the second coefficient. Recently, several authors have subsequently studied similar problems in this direction (see [2, 7]). For example, Brannan and Taha [1] considered certain subclasses of bi-univalent functions, similar to the familiar subclasses of univalent functions including of strongly starlike, starlike and convex functions. They introduced bi-starlike functions and bi-convex functions and obtained estimates on the initial coefficients.

In this section we introduce by Mσ(α) a certain subclass of bi-starlike functions as follows. Also, we obtain the bound for the initial coefficients.

Definition 3.1. A function fσ is said to be in the class Mσ(α), if the following inequalities hold:

1+απ2sinα<Rezf(z)f(z)<1+α2sinα  (zΔ).

and

1+απ2sinα<Rewg(w)g(w)<1+α2sinα  (wΔ),

where g(w)=f1(w) and π/2α<π.

For functions in the class Mσ(α), the following result is obtained.

Theorem 3.2.Let the function fA of the form (1.1) belongs to the class Mσ(α). Then

|a2|12+cosα  π/2α<π,

and

|a3|2+cosα  π/2α<π.

Proof. Let fMσ(α) and g=f1. Then using Lemma 1.2, there are analytic functions u,vB, satisfying

zf(z)/f(z)=1+Bα(u(z))andwg(w)/g(w)=1+Bα(v(z)),

where Bα(.) defined by (1.4). Define the functions k and l by

k(z)=1+u(z)1u(z)=1+k1z+k2z2+andl(z)=1+v(z)1v(z)=1+l1z+l2z2+,

or, equivalently,

u(z)=k(z)1k(z)+1=12k1z+k2k122 z2+,

and

v(z)=l(z)1l(z)+1=12l1z+l2l122 z2+.

It is clear that the functions k(z) and l(z)belong to class P and we have |ki|2 and |li|2(i=1,2,) (see [8]). However, clearly

zf(z)f(z)=1+Bαk(z)1k(z)+1andwg(w)g(w)=1+Bαl(z)1l(z)+1.

From (1.6), (3.6) and (3.7), we have

1+Bαk(z)1k(z)+1=1+12A1k1z+12A1 k2 k122+14A2k12z2+,

and

1+Bαl(z)1l(z)+1=1+12A1l1z+12A1 l2 l122+14A2l12z2+,

where A1=1 and A2=cosα, are given by (1.7). By suitably comparing coefficients of (3.5), we get

a2=12A1k1, 2a3a22=12A1k2k12 2+14A2k12, a2=12A1l1,

and

3a222a3=12A1l2l12 2+14A2l12.

From (3.11) and (3.13), we get

k1=l1

Also, from (3.12)-(3.15), we find that

a22=A13(k2+l2)4(A12+A1A2)=k2+l24(2+cosα)  (withA1=1andA2=-cosα).

Therefore, we have

|a22||k2|+|l2|4(2+cosα)12+cosα.

Thisgives the bound on |a2| as asserted in (3.3). Now, further computations from (3.12) and (3.14)-(3.16) lead to

a3=18A1(3k2+l2)+2k12(A2A1)=183k2+l2+2k12(cosα1).

Since |ki|2 and |li|2, we have

|a3|1+|1+cosα|.

Therefore, the proof of Theorem 3.2 is completed.

Corollary 3.3. Let the function f be in the class Mσ(π/2). Then

|a2|2/20.7071068,

and

|a3|2.

Also, if we take απ, in Theorem 3.2 we get

|ai|1  (i=2,3).

### References

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