### Article

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

**Published online** September 30, 2021

Copyright © Kyungpook Mathematical Journal.

### 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 Rzeszow, 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

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 *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(z^{k})]^{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

which are analytic and normalized by

A function

The equality

**Definition 1.1.** Let

Consider the function ϕ as follows

It is clear that

Thus, the class

By the subordination principle we have the following lemma.

**Lemma 1.2.** [4] Let

where

The function

in other words, the image of Δ is a vertical strip when

where

The following lemma will be useful.

**Lemma 1.3.** (see [9]) Let

then

This paper is organized as follows. In Section 2 we study the class

### 2. Coefficient Estimates

**Theorem 2.1.** ([10]) Let

Here, we considerthe problem of finding sharp upper bounds for the Fekete-Szegö coefficient functional associated with the

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

**Lemma 2.2.** [5] Let the function

be in the class

The result is sharp.

**Theorem 2.3.** Let

The result is sharp.

We define

and note that

where _{1}=1^{2}

and

Foreach

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

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

and

Therefore,

Applying Lemma 2.2 in (2.11) with

gives the inequality (2.3). For the sharpness it is sufficient to consider the

It is clear that

If we take in (2.12) _{1}=0_{2}=2

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

The problem of finding sharp upper bound for the coefficient functional

**Corollary 2.4.** Let

The result is sharp.

Putting

**Corollary 2.5.** Assume that the function

then

If we take

then

**Corollary 2.7.** Letthe function

and

We remark that every function

where

Applying Theorem 2.1 we get

The second inequality (2.17) follows by taking

### 3. Bi–Univalent Functions

First, we recall that a function ^{-1}

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

**Definition 3.1.** A function

and

where

For functions in the class

**Theorem 3.2.**Let the function

and

where

or, equivalently,

and

It is clear that the functions

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

and

where _{1}=1

and

From (3.11) and (3.13), we get

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

Therefore, we have

Thisgives the bound on

Since

Therefore, the proof of Theorem 3.2 is completed.

**Corollary 3.3.** Let the function

and

Also, if we take

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