Study on Certain Subclass of Analytic Functions Involving Mittag-Leffler Function

Abstract

We propose and explore a new subclass of regular functions described by a new derivative operator in this paper. Some coefficient estimations, growth and distortion aspects, extreme points, star-like radii, convexity, Fekete-Szego inequality, and partial sums are derived.

1. Introduction

Let represent the regular function class defined on the disk normalized by (i.e., and ). The origin of the form is about the Taylor series expansion of such an equation

indicates a subclass of consists entirely of mappings that are the same as .

For presented by (2) and specified by their convolution, represented by , is specified as

The subclass consisting of the -type function is specified by as

Silverman [1] extensively examined this subclass.

The study of operators plays an important role in geometric function theory in complex analysis and its related fields. Many derivative and integral operators can be written in terms of convolution of certain analytic functions. It is observed that this formalism brings an ease in further mathematical exploration and also helps to better understand the geometric properties of such operators. The Mittag-Leffler function [2, 3] is defined by the following power series, convergent in the whole complex plane:

We recognize that it is an entire function of order providing a simple generalization of the exponential function exp to which it reduces for For detailed information on the Mittag-Leffler-type functions and their laplace transforms, the reader may consult, e.g., [4–6] and the recent treatise by Gorenflo et al. [7].

We also note that for the convergence of the power series in (5), the parameter may be complex provided that The most interesting properties of the Mittag-Leffler function are associated with its asymptotic expansions as in various sectors of the complex plane. A more general function generalizing was introduced by Wiman [8] and defined by

Observe that the function contains many well-known functions as its special case, for example, , and

The Mittag-Leffler function arises naturally in the solution of fractional-order differential and integral equations and especially in the investigations of fractional generalization of kinetic equation, random walks, Levy flights, and super diffusive transport and in the study of complex systems. Several properties of Mittag-Leffler function and generalized Mittag-Leffler function can be found, e.g., in [9–16]. Observe that Mittag-Leffler function does not belong to the family Thus, it is natural to consider the following normalization of Mittag-Leffler functions as below:

It holds for complex parameters and

The function is specified by

Now, for , the derivative operator that follows is defined by by

If is specified by (1), then from the operator’s definition it is clear to see that where

Keep in mind that (1)the Al-Oboudi operator [17] is achieved when and (2)we get the Salagean operator [18] when , and (3)when , we get , according to Srivastava et al. [19]

If is represented by (4), then we have got it.

Now, by utilizing the differential operator, , a new subclass of functions belonging to the class is specified.

Definition 1. For , and , a mapping in a class is referred to as , if it satisfies the case where

We also define

For special situations of characteristics, and , it can be reduced to new or known categories of functions studied in recent research [20–25].

The objective of this review is to look into a variety of properties for functions in the aforesaid class. For specific parameter instances.

2. Coefficient Estimates

To get our results, we will require the subsequent lemma.

Lemma 2 (see [26]). Let be a real and be a complex number. Then, . For beginnings, we have a coefficient that is relevant for functions in the classs

Theorem 3. Let indicated by (1). If where then

Proof. In the definition by consequence of 1 and Lemma 2, it is enough to demonstrate that For the R.H.S and L.H.S of (18), we may, respectively, write and similarly, Then, The condition (16) required is fulfilled.

We have a necessary and adequate situation in the next theorem for a function to be in the class .

Theorem 4. Let indicated by (3). Then, . where is defined by (17).

Proof. We can only prove the requirement in view 3 of the theorem. If and is real, then We get the desired inequality from letting

Corollary 5. If , then

3. Growth and Distortion Theorem

Theorem 6. Let Then, for , where Equations (25) and (26) are sharp for the given function

Proof. Since and it follows from 4 of the theorem, where is given by (27), we have and therefore, Since is given by (3), we get In light of Theorem 4, we have which yields Thus, Hence, the proof is complete.

Consider that is sharp is to (25).

And is sharp is to (26).

4. Extreme Points

Now, for the function class, we look at the extreme points .

Theorem 7. Let the functions and Then, . where and

Proof. Assume that it is possible to write as in (37). Then, since By virtue 4 of the theorem, it follows that
Conversely, suppose and consider Then,, hence the theorem.

5. Radii of Starlikeness, Convexity, and Close-to-Convexity

Theorem 8. Let Then, is star-shaped of order in , where

Proof. To be able to prove the theorem, we have to demonstrate that for with We have Thus, In virtue of (22), we have The inequality of (43) would then be valid if or if Hence, the proof is complete.

The evidence 9 and 10 of the subsequent theorems is comparable to 8 of the theorem, so the evidence is excluded.

Theorem 9. Let Then, is convex of order in , where

Theorem 10. Let the function given by (3) be in the class . Then, in close-to-convex of order in , where

6. Fekete-Szego Inequality

In this section, for the mapping in the class, we get the Fekete-Szego inequality . To illustrate our fundamental result, we will identify the appropriate lemma.

Lemma 11 (see [27]). If is an analytic mapping with positive real part in , then When or , the inequality holds iff or one of its rotations. If , then the equality holds iff or one of its rotations. If , the equality holds iff or one of its rotations
If , the equality holds iff is the reciprocal of one of the mapping such that the equality holds when it comes to

Theorem 12. Let If is given by (1), then where

The outcome is sharp.

Proof. Since, for complex numbers, , implies that or that Hence, Let We then have, by way of (10) and (14), Therefore, we obtain We write where The implementation of the lemma above follows our conclusion. Denote If or , it is true that equality exists. When , it is true that equality exists, iff If , then it is true that equality exists, iff Finally, if it is true that equality exists , it is the inverse of one of the equality functions and holds true in the case of

7. Partial Sums

Consider the recent works on partial analytic function sums by Silverman [28] and Silvia [29]. Partial function in this class is considered in this section to be giving sharp lower boundaries to the reap part ratios of to and to

Theorem 13. Let and indicate and as partial sums Suppose that where Then,
Furthermore,

Proof. It is not crucial to verify that the coefficients supplied by (69) are correct. So we have The hypothesis used (69), by setting If we use and apply (73), we find that That immediately leads in a conclusion (70) of Theorem 13. To find out that gives sharp result, we observe that for, Similarly, if we take we can deduce, and make use of (73), that This leads directly to the statement (71) of Theorem 13.
For each with the external mapping , the bound in (71) is sharp indicated by (76).
Thus, the evidence of the Theorem 13 is complete.

Theorem 14. Let and fulfill (16). Then,

Proof. By setting Now, Now, since the L.H.S. of (83) is bounded above by if and the proof is complete.

The consequence of the extreme function is sharp

Theorem 15. Let and fulfill (16). Then,

Proof. By setting Using (84) and making use of it, we deduce that that immediately leads us to the statement 15 of the theorem.

8. Conclusions

This research has introduced study a new differential operator related to analytic function and studied some basic properties of geometric function theory. Accordingly, some results to coefficient estimates, grouth and distortion theorem, Fekete-Szego inequalityy, and partial sums have also been considered, inviting future research for this field of study.

Data Availability

No data were used to find this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

This work was equally contributed by all writers. The final version of the work has been read and approved by all of the authors.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.