Advances in Pure Mathematics
Vol.07 No.11(2017), Article ID:80546,6 pages
10.4236/apm.2017.711038
On a Subordination Result of a Subclass of Analytic Functions
Risikat Ayodeji Bello
Department of Mathematics and Statistics, College of Pure and Applied Science, Kwara State University, Malete, Nigeria
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: March 13, 2017; Accepted: November 21, 2017; Published: November 24, 2017
ABSTRACT
In this paper, we investigate a subordination property and the coefficient inequality for the class , The lower bound is also provided for the real part of functions belonging to the class .
Keywords:
Analytic Function, Univalent Function, Hadamard Product, Subordination
1. Introduction
Let A denote the class of function analytic in the open unit disk and let S be the subclass of A consisting of functions univalent in U and have the form
(1.1)
The class of convex functions of order in U, denoted as is given by
Definition 1.1. The Hadamard product or convolution of the func- tion and , where is as defined in (1.1) and the function is given by
is defined as:
(1.2)
Definition 1.2. Let and be analytic in the unit disk . Then is said to be subordination to in and written as:
if there exist a Schwarz function , analytic in U with , such that
(1.3)
In particular, if the function is univalent in U, then is said to be subordinate to if
(1.4)
Definition 1.3. The sequence of complex numbers is said to be a subordinating factor sequence of the function if whenever in the form (1.1) is analytic, univalent and convex in the unit disk , the subordination is given by
We have the following theorem:
Theorem 1.1. (Wilf [1] ) The sequence is a subordinating factor sequence if and only if
(1.5)
Definition 1.4. A function which is normalized by is said to be in if
The class was studied by Janwoski [2] . The family contains many interesting classes of functions. For example, for , if
Then is starlike of order in U and if
Then is convex of order in U.
Let be the subclass of consisting of functions such that
(1.6)
we have the following theorem
Theorem 1.2. [3] Let be given by Equation (1.6) with . If
then , .
It is natural to consider the class
Remark 1.1. [4] If , then consists of starlike functions of order , since
Our main focus in this work is to provide a subordination results for functions belonging to the class
2. Main Results
2.1. Theorem
Let , then
(2.1)
where and is convex function.
Proof:
Let
and suppose that
that is is a convex function of order .
By definition (1.1) we have
(2.2)
Hence, by Definition 1.3…to show subordination (2.1) is by establishing that
(2.3)
is a subordinating factor sequence with . By Theorem 1.1, it is sufficient to show that
(2.4)
Now,
Since ( ), therefore we obtain
which by Theorem 1.1 shows that is a subordinating factor, hence, we have established Equation (2.5).
2.2. Theorem
Given , then
(2.6)
The constant factor cannot be replaced by a larger one.
Proof:
Let
which is a convex function, Equation (2.1) becomes
Since
(2.7)
This implies
(2.8)
Therefore, we have
which is Equation (2.6).
Now to show that sharpness of the constant factor
We consider the function
(2.9)
Applying Equation (2.1) with and , we have
(2.10)
Using the fact that
(2.11)
We now show that the
(2.12)
we have
This implies that
and therefore
Hence, we have that
That is
which shows the Equation (2.12).
2.3. Theorem
Let
,
then .
Proof:
Let
then by definition of the class ,
we have that
which gives that
hence
Cite this paper
Bello, R.A. (2017) On a Subordination Result of a Subclass of Analytic Functions. Advances in Pure Mathematics, 7, 641-646. https://doi.org/10.4236/apm.2017.711038
References
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