Journal of Function Spaces
Volume 2021 (2021), Article ID 6739237, 7 pages
https://doi.org/10.1155/2021/6739237

Research Article

On Some Relationships of Certain Uniformly Analytic Functions Associated with Mittag-Leffler Function

Adel A. Attiya,1,2 Ekram E. Ali,1,3 Taher S. Hassan,1,2 and Abeer M. Albalahi¹

¹Department of Mathematics, College of Science, University of Ha’il, Ha’il 81451, Saudi Arabia
²Department of Mathematics Faculty of Science, Mansoura University, Mansoura 35516, Egypt
³Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt

Correspondence should be addressed to Adel A. Attiya; aattiy@mans.edu.eg

Received 14 April 2021; Accepted 30 April 2021; Published 28 May 2021

Academic Editor: Gangadharan Murugusundaramoorthy

Copyright © 2021 Adel A. Attiya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we introduce and investigate several inclusion relationships of new -uniformly classes of analytic functions defined by the Mittag-Leffler function. Also, integral-preserving properties of these classes associated with the certain integral operator are also obtained.

1. Introduction

Let be the class of analytic functions in the open unit disc which in the form

For and , we say that the function is subordinate to , written symbolically as follows:

if there exists a Schwarz function , which (by definition) is analytic in with and , , such that for all . In particular, if the function is univalent in , then we have the following equivalence relation (cf., e.g., [1, 2]; see also [3]):

Let be as in (1) and then Hadamard product (or convolution) of and is given by

For , we denote by and the subclasses of consisting of all analytic functions which are, respectively, starlike of order , convex of order , close-to-convex of order and type and quasiconvex of order and type in .

Also, let the subclasses , , and of be defined as follows:

We note that

Moreover, let be an analytic function which maps onto the conic domain such that defined as follows:

where and is such that By virtue of properties of the conic domain (cf., e.g., [4, 5]), we have

Making use of the principal of subordination and the definition of , we may rewrite the subclasses , , , and as follows:

and

Attiya [6] introduced the operator , where is defined by

with and . Also, when ; Here, is the generalized Mittag–Leffler function defined by [7], see also [6], and the symbol () denotes the Hadamard product.

Due to the importance of the Mittag–Leffler function, it is involved in many problems in natural and applied science. A detailed investigation of the Mittag–Leffler function has been studied by many authors (see, e.g., [7–12]).

Attiya [6] noted that

Also, Attiya [6] showed that

and

Next, by using the operator , we introduce the following subclasses of analytic functions in

where and . Also, when ;

Also, we note that

In this paper, we introduce several inclusion properties of the classes , , , and Also, integral-preserving properties of these classes associated with generalized Libera integral operator are also obtained.

2. Inclusion Properties Associated with

Lemma 1 (see [13]). If is convex univalent in with and . Let be analytic in with which satisfy the following subordination relation

then

Lemma 2 (see [2]). If is convex univalent in and let be analytic in with Let be analytic in and which satisfy the following subordination relation

then

Theorem 3. If then

Proof. Let put

we note that is analytic in and . From (13) and (22), we have

Differentiating (23) with respect to , we obtain

From the above relation and using (7), we may write

Since , we see that

Applying Lemma 1, it follows that , that is, .

Using the same technique in Theorem 3 with relation (14), we have the following theorem.

Theorem 4. If then

Theorem 5. If , then .

Proof. Applying Theorem 3 and relation (16), we observe that

which evidently proves Theorem 5.

Similarly, we can prove the following theorem.

Theorem 6. If then

Theorem 7. If , then

Proof. Let . Then, there exists a function such that

We can choose the function such that . Then, and

Now, let

where is analytic in with . Since , by Theorem 3, we know that . Let

where is analytic in with Also, from(30), we note that

Differentiating both sides of (32) with respect to , we obtain

Now, using (13) and (33), we obtain

Since we see that

Hence, applying Lemma 2, we can show that so that . This completes the proof of Theorem 7.

Similarly, we can prove the following theorem.

Theorem 8. If , then

We can also prove Theorem 9 by using Theorem 7 and relation (17).

Theorem 9. If , then

Also, we obtain the following theorem.

Theorem 10. If , then

Now, we obtain squeeze theorems for inclusion by combining the above theorems as follows:

Combining both theorems 3 and 4, we have the following corollary.

Corollary 11. If then

Combining both theorems 5 and 6, we have the following corollary.

Corollary 12. If then

Combining both theorems 7 and 8, we have the following corollary.

Corollary 13. If then

Combining both theorems 9 and 10, we have the following corollary.

Corollary 14. If then

3. Integral Preserving Properties Associated with

The generalized Libera integral operator (see [14–16], also, see related topics [17–19]) is defined by

where and

Theorem 15. Let . If , then

Proof. Let and set

where is analytic in with . From definition of and (40), we have

Then, by using (41) and (42), we obtain

Taking the logarithmic differentiation on both sides of (43) and simple calculations, we have

Since by virtue of Lemma 1, we conclude that in , which implies that .

Theorem 16. Let . If , then .

Proof. By applying Theorem 15, it follows that

which proves Theorem 16.

Theorem 17. Let . If , then .

Proof. Let . Then, there exists a function such that

Thus, we set

where is analytic in with . Since , we see from Theorem 15 that . Let

where is analytic in with . Using (47), we have

Differentiating both sides of (49) with respect to and simple calculations, we obtain

Now, using the identity (42) and (50), we obtain

Since and , we see that

Applying Lemma 2 into relation (51), it follows that which is .

We can deduce the integral-preserving property asserted by 18 by using Theorem 17 and relation (17).

Theorem 18. Let . If , then .

Data Availability

All data are available in this paper.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

The authors contributed equally to the writing of this paper. All authors approved the final version of the manuscript.

Acknowledgments

This research has been funded by Scientific Research Deanship at the University of Ha'il, Saudi Arabia, through project number RG-20020.

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