New Subclass of Analytic Function Involving -Mittag-Leffler Function in Conic Domains

Abstract

In this paper, we formulate the -analogus of differential operator associated with -Mittag-Leffler function. By using this newly defined operator, we define a new subclass of analytic functions in conic domains. We investigate the number of useful properties such that structural formula, coefficient estimates, Fekete–Szego problem and subordination result. We also highlighted some known corollaries of our main results.

1. Introduction Definition

Let denote the class of functions which are analytic in the open unit disk , satisfying the condition and , and for every has the series expansion of the form

Let be the class of all functions which are univalent in (see [1]). Also, denotes the well-known Carathéodory class of functions which are analytic in open unit disk and has the series expansion of the form and satisfying the condition

For the function given by (1) and the function defined by the Hadamard product (convolution) of the functions and stated by

For the analytic functions , is said to be subordinate to (indicated as , if there exists a Schwarz function with such that

Furthermore, if is univalent in , (see [2]); then, we have

The class of starlike functions of order in and the class of convex functions of order , , were defined as follows:

It should be noted that where and are the well-known function classes of starlike and convex functions, respectively.

In the year of 1991, Goodman [3] introduced the class of uniformly convex functions which was extensively studied by Ronning [4], and its characterization was given by Ma and Minda [5]. After that, Kanas and Wisniowska [6] defined the class -uniformly convex functions (- and a related class was defined by

From different viewpoints, the various subclasses of the normalized analytic function of class have been studied in the field of Geometric Function Theory. To investigate various subclasses of , many authors have been used the -calculus as well as the fractional -calculus. In 1910, Jackson [7] was among the one of few researchers who studied -calculus operator theory on -definite integrals and also Trjitzinsky in [8] studied about analytic theory of linear -difference equations. Curmicheal [9] studied general theory of linear -difference equations and the first use of -calculus operator theory in Geometric Function Theory in a book chapter by Srivastava (see, for details, [10]). Recently, Hussain et al. discussed the some applications of -calculus operator theory in [11], while in [12, 13], Ibrahim et al. used the notion of quantum calculus and the Hadamard product to improve an extended Sàlàgean -differential operator and defined some new subclasses of analytic functions in open unit disk . Govindaraj and Sivasubramanian [14] as well as Ibrahim et al. [15, 16] employed the quantum calculus and the Hadamard product to defined some new subclasses of analytic functions involving the Sàlàgean -differential operator and the generalized symmetric Sàlàgean -differential operator, respectively. Furthermore, Srivastava et al. [17] defined -Noor integral operator by using -calculus operator theory and investigated some subclasses of biunivalent functions in open unit disk.

Here, we give some basic definitions and details of the -calculus and suppose that

For any nonnegative integer , the -integer number is defined by and for any nonnegative integer , the -number shift factorial is defined by

We note that when then .

The -difference operator was introduced by Jackson (see in [7]). For the -derivative operator or -difference operator is defined as

It is readily deduced from (1) and (14) that

We can observe that

The familiar Mittag-Leffler function introduced by Mittag-Leffler [18] and its generalization introduced by Wiman [19] which are defined by

Recently, Attiya [20] investigated some applications of Mittag-Leffler functions and generalized -Mittag-Leffler studied by Rehman et al. in [21]. Moreover, Srivastava et al. [22, 23] introduced the generalization of Mittag-Leffler functions.

The -Mittag-Leffler function was defined by (see [24]):

The -Mittag-Leffler function has also been investigated in [25, 26]. Since the -Mittag-Leffler function defined by (18) does not belong to the normalized analytic function class . Hence, we define the normalization of -Mittag-Leffler function as where , Corresponding to and for , we define the following -analogous of differential operator by

We note that where

Note that (i)For and we get Salagean -differential operator [14](ii)For and we get Salagean differential operator [27](iii)For we get (see [24])(iv)For we get (see [22])

Definition 1. Let , then is in the class , if it satisfies the condition

Remark 2. (i)For and , the class studied in [11](ii)For and the class studied in [28](iii)For and the class studied in [29](iv)For and the class studied in [30](v)For and studied in [30]

2. Geometric Interpretation

A function belongs to if and only if takes all the values in the conic domain , such that where

The domain is not always well defined because in general (for example, in particular ). We see that in [31], the conic domain concides with only when is chosen according to (i)For we take (ii)For , we take (iii)For , we take (iv)For , we take

This means that for to contain the point must be chosen according as follows:

Since is convex univalent, the above definition can be written as where

For more detail (see [32, 33]).

3. Set of Lemmas

Lemma 3. (see [34]). Let in If is convex univalent in , then

Lemma 4. (see [35]). Let be fixed and let of the form (28). If where

Lemma 5. (see [36]). Let and let be analytic in and satisfy for in , then

4. Main Results

Theorem 6. Let Then, where is a Schwarz function given in (5) Moreover, for we have where is given by (28).

Proof. If then by using (27), we obtain Integrating (39) and after some simplification, we have This proves (37). We know that Using (40) and (41), we have for . From (40), we have which implies that

Corollary 7. (see [11]). Let Then, where is a Schwarz function given in (5) Moreover, for we have

Theorem 8. If Then, where with and are given by (34) and (22).

Proof. Let where is the analytic in and Let and is given by (21). Then, (49) implies that Now comparing the coefficients of , we obtain which implies Using the results that given in ([33]), we have Let us take Then, we have For in (54), we have Hence, for the inequality (48) holds. To prove (48), we use mathematical induction, for By using (55), we have Therefore, Hence, (48) holds for . Now, we suppose that (48) is true for , that is Consider Hence, (48) holds for Hence, proof is complete.

Corollary 9. (see [11]). f Then,

Theorem 10. Let be fixed and let with the form (1), then, for where and are given by (34) and (35).

Proof. Let then there exists a Schwarz function given by (5), such that Let be defined as This gives Using (67) in (64) and comparing with (68), we obtain For any complex number and after some calculation we have where Using a lemma (36) on (70), we have the required result.

Corollary 11. (see [11]). Let be fixed and let with the form (1.1), then, for where

Theorem 12. Let of the form (1) and satisfy the condition then,

Proof. Let we note that From (74), we get To show that , it suffices that From (Proof), we have Because of (74).

Corollary 13. (see [11]). If a function of the form (1) and satisfy the condition then,

Corollary 14. (see [28]). A function of the form (1) belongs to , if where and . Then,
When then, with and .

Corollary 15. (see [37]). A function of the form (1) is in the class , if

Theorem 16. Let Then, includes an open disk of radius where is given by (34).

Proof. Let a nonzero complex number such that for Then, Since is univalent, therefore Now using (47), we have Hence we have When and then we have known result [11].

Corollary 17. Let Then, includes an open disk of radius

5. Conclusion

In this paper, we formulate the -analogous of differential operator associated with -Mittag-Leffler function. By applying newly defined operator, we defined and investigated a new subclass of analytic functions in conic domains. We investigated the number of useful properties such that structural formula, coefficient estimates, Fekete–Szego problem, and subordination results. We also highlighted some known consequences of our main result. For future work, one can employ the -analogous of differential operator (21) in different classes of analytic functions such as the meromorphic and multivalent functions (see [38–42].

Data Availability

All data are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was sponsored by the Deanship of Scientific Research under Nasher Proposal No. 216106, King Faisal University, Al-Ahsa, Hofuf, Saudi Arabia.