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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+α−π2sinα 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

$f(z)=z+∑ n=2∞anzn,$

which are analytic and normalized by $f(0)=f′(0)−1=0$ in the open unit disk $Δ={z∈ℂ:|z|<1}$. 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 $|w(z)|<1$, $z∈Δ$. 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 $w∈B$ 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:

$f(z)≺g(z)⇔(f(0)=g(0) and f(Δ)⊂g(Δ)).$

A function $f∈S$ is starlike (with respect to 0) if $tw∈f(Δ)$ whenever $w∈f(Δ)$ and $t∈[0,1]$. The class of starlike functions is denoted by $S*$. We say that $f∈S*(γ)$ ($0≤γ<1$) if and only if

$Rezf′(z)f(z)>γ (z∈Δ).$

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 $π/2≤α<π$. Then the function $f∈A$ belongs to the class $M(α)$ if f satisfies

$1+α−π2sinα

Consider the function ϕ as follows

$ϕ(α):=1+α−π2sinα (π/2≤α<π).$

It is clear that $ϕ(π/2)=1−π/4≈0.2146$ and

$limα→π−ϕ(α)=12.$

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

By the subordination principle we have the following lemma.

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

$zf′(z)f(z)−1≺Bα(z) (z∈Δ),$

where

$Bα(z):=12isinαlog1+zeiα1+ze−iα (z∈Δ).$

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

$Ωα:=w:α−π2sinα

in other words, the image of Δ is a vertical strip when $π/2≤α<π$. 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

$Bα(z)=∑ n=1∞Anzn (z∈Δ),$

where

$An=(−1)(n−1)einα−e−inα2insinα (n=1,2,…).$

The following lemma will be useful.

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

$p(z)≺q(z) (z∈Δ),$

then

$|Pn|≤|Q1| n≥1.$

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 $π/2≤α<π$. If a function $f∈A$ 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

$F(z)=[f(zk)]1/k=z+∑ n=1∞bkn+1zkn+1 (z∈Δ).$

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

$|c2−μc12|≤2max{1,|2μ−1|}.$

The result is sharp.

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

$b2k+1−μbk+12≤12kmax1,2μ−k−1+kcosα2k.$

The result is sharp.

Proof. Since $f∈M(α)$, from Lemma 1.2 and by definition of subordination, there exists a function $w∈B$ such that

$zf′(z)/f(z)=1+Bα(w(z)).$

We define

$p(z):=1+w(z)1−w(z)=1+p1z+p2z2+⋯,$

and note that $p∈P$. 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

$a3=18(1−cosα)p12+14p2−12p12.$

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

$F(z)=[f(z1/k)]1/k=z+1ka2zk+1+1ka3−12k−1k2a22z2k+1+⋯.$

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

$bk+1=1ka2 and b2k+1=1ka3−12k−1k2a22.$

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

$bk+1=p12k,$

and

$b2k+1=18k1−cosα−k−1kp12+14kp2−12p12.$

Therefore,

$b2k+1−μbk+12=14kp2−2μ+k−1+kcosα2kp12.$

Applying Lemma 2.2 in (2.11) with

$μ′=2μ+k−1+kcosα2k,$

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

$f(z)=zexp∫0z Bα(w(t))tdt.$

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

$b2k+1−μbk+12=12k2μ−k−1+kcosα2k.$

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

$b2k+1−μbk+12=12k.$

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

The problem of finding sharp upper bound for the coefficient functional $|a3−μa22|$ 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 $f∈M(α)$. Then, for any complex number µ,

$a3−μa22≤12max1,2μ−2+cosα2.$

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:

$1−π4

then

$a3−μa22≤12max1,μ−1 (μ∈ℂ).$

If we take $α→π−$ 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:

$Rezf′(z)f(z)>1−π4 z∈Δ,$

then

$a3−μa22≤12max1,(2μ−3)/2 (μ∈ℂ).$

Corollary 2.7. Letthe function f, given by (1.1), be in the class $M(α)$. Also let the function $f−1(w)=w+∑ n=2∞bnwn$ be the inverse of f. Then

$|b2|≤1,$

and

$|b3|≤12|6−cosα| π/2≤α<π.$

We remark that every function $f∈S$ has an inverse $f−1$, defined by $f−1(f(z))=z$ ($z∈Δ$) and

$f(f−1(w))=w (|w|

where

$f−1(w)=w−a2w2+(2a22−a3)w3−(5a23−5a2a3+a4)w4+⋯.$

Proof Comparing (2.18) with $f−1(w)=w+∑ n=2∞bnwn$, gives us

$b2=−a2 and b3=2a22−a3.$

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 $f∈A$ is said to be bi-univalent in Δ if f univalent in Δ and f-1 has an univalent extension from $|w| 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 $f∈σ$ is said to be in the class $Mσ(α)$, if the following inequalities hold:

$1+α−π2sinα

and

$1+α−π2sinα

where $g(w)=f−1(w)$ and $π/2≤α<π$.

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

Theorem 3.2.Let the function $f∈A$ of the form (1.1) belongs to the class $Mσ(α)$. Then

$|a2|≤12+cosα π/2≤α<π,$

and

$|a3|≤2+cosα π/2≤α<π.$

Proof. Let $f∈Mσ(α)$ and $g=f−1$. Then using Lemma 1.2, there are analytic functions $u,v∈B$, satisfying

$zf′(z)/f(z)=1+Bα(u(z)) and wg′(w)/g(w)=1+Bα(v(z)),$

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

$k(z)=1+u(z)1−u(z)=1+k1z+k2z2+⋯ and l(z)=1+v(z)1−v(z)=1+l1z+l2z2+⋯,$

or, equivalently,

$u(z)=k(z)−1k(z)+1=12k1z+k2−k122 z2+⋯,$

and

$v(z)=l(z)−1l(z)+1=12l1z+l2−l122 z2+⋯.$

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

$zf′(z)f(z)=1+Bαk(z)−1k(z)+1 and wg′(w)g(w)=1+Bαl(z)−1l(z)+1.$

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

$a2=12A1k1,$ $2a3−a22=12A1k2−k12 2+14A2k12,$ $−a2=12A1l1,$

and

$3a22−2a3=12A1l2−l12 2+14A2l12.$

From (3.11) and (3.13), we get

$k1=−l1$

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

$a22=A13(k2+l2)4(A12+A1−A2)=k2+l24(2+cosα) (with A1=1 and A2=-cosα).$

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

$a3=18A1(3k2+l2)+2k12(A2−A1)=183k2+l2+2k12(−cosα−1).$

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

$|a2|≤2/2≈0.7071068…,$

and

$|a3|≤2.$

Also, if we take $α→π−$, in Theorem 3.2 we get

$|ai|≤1 (i=2,3).$

### References

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