#### 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.