### Article

Kyungpook Mathematical Journal 2023; 63(2): 225-234

**Published online** June 30, 2023 https://doi.org/10.5666/KMJ.2023.63.2.225

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 : ama19u2001@uoanbar.edu.iq and eps.abdulrahman.juma@uoanbar.edu.iq

Department of Mathematics, Faculty of Arts and Sciences, Bursa Uludag University, 16059, Görükle, Bursa, Turkey

e-mail : syalcin@uludag.edu.tr

**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

### 1. Introduction

Let

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

for

The operator

(i)

(ii)

(iii)

(iv)

(v)

(vi)

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

for

The following lemmas will be useful in deriving our results.

**Lemma 1.2.** ([8]) If

then the differential equation

has a univalent solution given by

If

then

**Lemma 1.3.**([7]) Let

(i)

(ii)

If

then

and

### 2. Main Results

For

**Theorem 2.1.** Let

where

and

Further,

then

then differentiating (2.5) with respect to

Applying Lemma 1.2 for

where

**Theorem 2.2.** If

such that

and

Further,

Thus, the function

then differentiating (2.10) with respect to

By make use of Lemma 1.2 for

where

**Theorem 2.3.** If

where

and

Further,

Subsequently,

By differentiation (2.15) with respect to

Thus, from Lemma 1.2 for

where

**Theorem 2.4.** Suppose that

Let

where

then

for

Since

Hence, both (i) and (ii) of Lemma 1.3 are satisfied. Consider the function

subsequently, the function

equivalent to

Thus, applying Lemma 1.3 it follows that

From Theorem 2.4, we obtain the following result.

**Corollary 2.5.** Let

If

then

with

We took

**Corollary 2.6.** Let

If

then

For

Taking

**Corollary 2.7.** Suppose that

If

then

For

We took

**Corollary 2.8.** Let

If

then

For

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.

- F. M. Al-Oboudi,
On univalent functions defined by a generalized Sălăgean operator , Int. J. Math. Math. Sci.,27 (2004), 1429-1436. - A. Catas,
On a certain differential sandwich theorem associated with a new generalized derivative operator , Gen. Math.,17(4) (2009), 83-95. - P. L. Duren. Univalent functions. Springer Science & Business Media; 2001.
- C. Y. Gao, S. M. Yuan and H. M. Srivastava,
Some functional inequalities and inclusion relationships associated with certain families of integral operators , Comput. Math. Appl.,49(11-12) (2005), 1787-1795. - I. B. Jung, Y. C. Kim and H. M. Srivastava,
The Hardy space of analytic functions associated with certain one-parameter families of integral operators , J. Math. Anal. Appl.,176(1) (1993), 138-147. - Y. Komatu,
On analytic prolongation of a family of operators , Mathematica,32(55) (1990), 141-145. - S. S. Miller and P. T. Mocanu. Differential subordinations: theory and applications. CRC Press; 2000.
- J. Patel, N. E. Cho and H. M. Srivastava,
Certain subclasses of multivalent functions associated with a family of linear operators , Math. Comput. Modelling,43(3-4) (2006), 320-338. - R. K. Raina and P. Sharma,
Subordination properties of univalent functions involving a new class of operators , Electron. J. Math. Anal. Appl.,2(1) (2014), 37-52. - G. S. Salagean. Subclasses of univalent functions. Berlin: Springer; 1983.
- C. Selvaraj and K. R. Karthikeyan,
Differential subordination and superordination for analytic functions defined using a family of generalized differential operators , An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. ,17(1) (2009), 201-210.