Article
KYUNGPOOK Math. J. 2019; 59(2): 215-224
Published online June 23, 2019
Copyright © Kyungpook Mathematical Journal.
Coefficient Bounds for Several Subclasses of Analytic and Biunivalent Functions
Samira Rahrovi∗ and Hossein Piri, Janusz SokÓŁ
Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran
e-mail : s.rahrovi@bonabu.ac.ir and hossein_piri1979@yahoo.com
Faculty of Mathematics and Natural Sciences, University of Rzeszów, Prof. Pigonia street 1, 35-310 Rzeszów, Poland
e-mail : jsokol@ur.edu.pl
Received: December 17, 2016; Revised: March 20, 2019; Accepted: April 23, 2109
In the present paper, some generalizations of analytic functions have been considered and the bounds of the coefficients of these classes of bi-univalent functions have been investigated.
Keywords: bi-univalent function, starlike function of order α, strongly starlike functions, strongly convex functions of order α, subordination.
1. Introduction and Preliminaries
Let denote the class of functions of the form
which are analytic in the open unit disk
then
A function
and is said to be in the class of strongly convex functions of order
For two functions
For each
and
A function is said to be bi-univalent in
Let ∑ denote the class of all bi-univalent functions defined in the unit disk
Lewin [8] considered the class of bi-univalent functions ∑ and obtained the following bound for the second coefficient in the Tylor-Maclaurin expansion as follows
for function
is presumably still an open problem. The bi-univalent functions was actually revived in the recent years by the pioneering work by Srivastava et al. [15]. In [15], two new subclasses of the function class ∑ has been introduced and the estimates on the coefficients |
We notice that the class ∑ is not empty. For example, the following functions are members of ∑:
However, the Koebe function is not a member of ∑.
In the next section, motivated essentially by the recent work of Srivastava et al. [15], Frasin and Aouf [6], Xu et al. [16] and other authors, we investigte interesting subclasses of analytic and bi-univalent functions in the open unit dick
2. Coefficient Estimates
In the sequel, it is assumed that
Suppose that
It is well known that
By a simple calculation, we have
Definition 2.1
A function
and
where
For special choices for the number
For proving our results we used the subordination concept which have been used by many authors (for example see [1]).
Theorem 2.2
Let
and
Since
and
it follows from (
From (
and
By adding the relation (
Now, in view of (
Then, we get
Next, from (
Then in view of (
It follows from (
Now from (
This completes the proof.
Remark 2.3
Let
then inequalities (
and
Definition 2.4
A function
and
where
Theorem 2.5
Let
and
Since
and
it follows from (
From (
and
By adding the relation (
Now, in view of (
Then, we get
Next, from (
Then in view of (
It follows from (
Now from (
This completes the proof.
Remark 2.6
If let
in theorem 2.5, then inequalities (
and
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