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Developments in Geometric Function Theory

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Volume 2021 |Article ID 6739237 | https://doi.org/10.1155/2021/6739237

Adel A. Attiya, Ekram E. Ali, Taher S. Hassan, Abeer M. Albalahi, "On Some Relationships of Certain Uniformly Analytic Functions Associated with Mittag-Leffler Function", Journal of Function Spaces, vol. 2021, Article ID 6739237, 7 pages, 2021. https://doi.org/10.1155/2021/6739237

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

Academic Editor: Gangadharan Murugusundaramoorthy
Received14 Apr 2021
Accepted30 Apr 2021
Published28 May 2021

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., [712]).

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 [1416], also, see related topics [1719]) 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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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.

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