Article
KYUNGPOOK Math. J. 2019; 59(4): 651-663
Published online December 23, 2019
Copyright © Kyungpook Mathematical Journal.
On A Subclass of Harmonic Multivalent Functions Defined by a Certain Linear Operator
Hanan Elsayed Darwish, Abdel Moneim Yousof Lashin and Suliman Mohammed Sowileh∗
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
e-mail : Darwish333@yahoo.com, aylashin@mans.edu.eg and s_soileh@yahoo.com
Received: November 7, 2016; Revised: February 24, 2018; Accepted: March 9, 2018
Abstract
In this paper, we introduce and study a new subclass of p-valent harmonic functions defined by modified operator and obtain the basic properties such as coefficient characterization, distortion properties, extreme points, convolution properties, convex combination and also we apply integral operator for this class.
Keywords: harmonic, multivalent functions, distortion bounds, extreme points.
1. Introduction
Harmonic mappings have found several applications in many diverse fields such as operations research, engineering, and other allied branches of applied mathematics. A continous function
where
If
The class ℋ1 = ℋ of harmonic univalent functions studied by Jahangiri et al. [10] (see also [6], [12]). For complex parameters
where
and (
For 1
Let
We note that by the special choices of
(i) For
where
(ii) For
where
(iii) For
where
(iv) For
where
(v) For
where
2. Coefficient Characterization
Unless otherwise mentioned, we assume throughout this article that 1
Theorem 2.1
Using the fact that ℜ{
We have
which is bounded above by 1 by using (
Theorem 2.2
Since
is equivalent to
Letting
Remark 2.1
(i) If
(ii) If
3. Extreme Points and Distortion Theorem
In the following theorem we give the extreme points of the closed convex hulls of the class
Theorem 3.1
Suppose that
Then
and so
Conversely, if
and
Then note that by Theorem 2.2, 0 ≤
The required representation is obtained as
This completes the proof of Theorem 3.1.
The following theorem gives the distortion bounds for functions in the class
Theorem 3.2
Let
Similarly we can prove
Remark 2.2
The bounds given in Theorem 3.2 for functions
and
showing that the bounds given in Theorem 3.2 are sharp.
4. Closure Property of the Class H p , q , s - ( n , ℓ , λ , α 1 , γ )
In the next two theorems, we prove that the class
and
is defined as
Using this definition, the next theorem shows that the class
Theorem 4.1
Let the functions
since 1
Now, we show that the class
Theorem 4.2
For i = 1, 2, 3, …, suppose
Then by (
For
Using the inequality (
which is the required coefficient condition.
Finally, we examine the closure property of the class
Theorem 4.3
From the representation of
where
Therefore, we have
Hence by Theorem 2.2,
Acknowledgements
The authors wish to acknowledge and thank both the reviewers and editors for job well-done reviewing this article.
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