Kyungpook Mathematical Journal 2023; 63(2): 225-234
Published online June 30, 2023
Copyright © Kyungpook Mathematical Journal.
Subordination Properties for Classes of Analytic Univalent Involving Linear Operator
Amal Madhi Rashid, Abdul Rahman S. Juma, Sibel Yalçın∗
Department of Mathematics, College of Education for Pure Sciences, University of Anbar, Ramadi, Iraq
e-mail : firstname.lastname@example.org and email@example.com
Department of Mathematics, Faculty of Arts and Sciences, Bursa Uludag University, 16059, Görükle, Bursa, Turkey
e-mail : firstname.lastname@example.org
Received: July 25, 2022; Revised: March 23, 2023; Accepted: March 29, 2023
In this paper, we use the use the linear operator
Keywords: Analytic function, Univalent function, Differential subordination, Starlike function
in the open unit disk
The Hadamard product (or convolution)
is given by
For two functions
if there exists a Schwarz function
Now, for function
From (1.5), it is easy to show that
Using the concept of subordination in (1.4) and the operator
Definition 1.1. A function
The following lemmas will be useful in deriving our results.
Lemma 1.2. () If
then the differential equation
has a univalent solution given by
Lemma 1.3.() Let
Theorem 2.1. Let
then differentiating (2.5) with respect to
Applying Lemma 1.2 for
Theorem 2.2. If
then differentiating (2.10) with respect to
By make use of Lemma 1.2 for
Theorem 2.3. If
By differentiation (2.15) with respect to
Thus, from Lemma 1.2 for
Theorem 2.4. Suppose that
Hence, both (i) and (ii) of Lemma 1.3 are satisfied. Consider the function
subsequently, the function
Thus, applying Lemma 1.3 it follows that
From Theorem 2.4, we obtain the following result.
Corollary 2.5. Let
Corollary 2.6. Let
Corollary 2.7. Suppose that
Corollary 2.8. Let
In the present work, we were able to obtain the best results, or best dominants of the subordination. Our main results give an interesting process for the study of many analytic univalent classes earlier defined by several authors. These classes expand and generalize many of those defined by many specialists in this field. Furthermore, the general subordination theorems lead us to some special cases that were used to determine new results connected with the classes we investigated.
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