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On Some Relationships of Certain Uniformly Analytic Functions Associated with Mittag-Leffler Function
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.
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 ):
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:
Attiya  introduced the operator , where is defined by with and . Also, when ; Here, is the generalized Mittag–Leffler function defined by , see also , 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  noted that
Also, Attiya  showed that
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 ). If is convex univalent in with and . Let be analytic in with which satisfy the following subordination relation then
Lemma 2 (see ). 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 .
All data are available in this paper.
Conflicts of Interest
The authors declare no conflict of interest.
The authors contributed equally to the writing of this paper. All authors approved the final version of the manuscript.
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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