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 : firstname.lastname@example.org
College of Natural Sciences, University of Rzeszow, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
e-mail : email@example.com
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
Keywords: analytic functions, starlike and bistarlike functions, subordination, Fekete-Szegö, inequality.
which are analytic and normalized by
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.  Let
in other words, the image of Δ is a vertical strip when
The following lemma will be useful.
Lemma 1.3. (see ) Let
This paper is organized as follows. In Section 2 we study the class
2. Coefficient Estimates
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 .
Lemma 2.2.  Let the function
be in the class
The result is sharp.
Theorem 2.3. Let
The result is sharp.
Applying Lemma 2.2 in (2.11) with
gives the inequality (2.3). For the sharpness it is sufficient to consider the
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.
Corollary 2.5. Assume that the function
If we take
Corollary 2.7. Letthe function
We remark that every function
Applying Theorem 2.1 we get
The second inequality (2.17) follows by taking
3. Bi–Univalent Functions
First, we recall that a function
In 1967 Lewin  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  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
For functions in the class
Theorem 3.2.Let the function
It is clear that the functions
From (3.11) and (3.13), we get
Therefore, we have
Therefore, the proof of Theorem 3.2 is completed.
Corollary 3.3. Let the function
Also, if we take
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