Abstract

In the present paper, we investigate and introduce several properties of certain families of analytic functions in the open unit disc, which are defined by -analogue of Mittag-Leffler function associated with conic domain. A number of coefficient estimates of the functions in these classes have been obtained. Sufficient conditions for the functions belong to these classes are also considered.

1. Introduction

Denoted by , the class of all analytic functions in has the following form:

For two functions and , analytic in , we say that the function is subordinate to , written as , if there exists a Schwarz function , which is analytic in with and such that . Furthermore, if the function is univalent in , then we have the following equivalence:

Let be the subclass of consisting of univalent functions. Also, let and denote the subclasses of univalent starlike and convex functions of order (). A function is said to be -starlike function, written as , if

A function is said to be -convex function, written as , if

The classes and were introduced and studied by Kanas and Wiśniowska [1, 2] (see also, [3–8]). In particular, when , we get and , where and are the familiar classes of uniformly starlike functions and uniformly convex functions in , respectively (see [9]). A function is said to be in the class that are -starlike functions of order with if

A function is said to be in the class that are -convex functions of order with if

The classes and have been studied by Kanas and Rducanu [10]. We note that and . The classes and were investigated in [11–13].

The study of quantum calculus (or -calculus) attracted the researches due to its applications in various branches of mathematics and physics, for example, in the areas of special functions, -difference, ordinary fractional calculus, -integral equations, and in -transform analysis (see [14–23]).

For and given by (1), the -derivative of is defined by (see [24–29]): provided that exists. From (1) and (7), we have where is -integer number defined by

We note that for a function which is differentiable in . Making use of the -derivative operator given by (7), we introduce the subclasses and in as follows.

Definition 1. For , , , and , let and be the subclasses of consisting of functions of the form (1) and satisfy the analytic criterion:

From (11) and (12), it follows that

The -shifted factorials, for any complex number , are defined by

The definition (14) remains meaningful for as a convergent infinite product

Furthermore, in terms of the -gamma function defined by so that for the familiar gamma function , we find from (14) that

We note that where

For , , , , and , consider the -analogue of Mittag-Leffler defined by Sharma and Jain [30], for generalized Mittag-Leffler function (see, e.g., [31–33])

As , the operator reduces to introduced by Prabhakar [34]. Now, let us define

We remark that: (i)(ii)where is one of the -analogues of the exponential function given by

Using the Hadamard product (or convolution), we define the linear operator by

We note that (see [35, 36])

Motivated by the works of Kanas and Yaguchi [37] and Kanas and Rducanu [10], we define the following classes of functions with the theory of -calculus.

Definition 2. For , , , , , , and , let It is easy to check that

Taking in Definition 2, we obtain

Motivated by the works mentioned above, in this paper, we will investigate some important properties, coefficient estimates, and the familiar Fekete–Szegö type inequalities for the subclasses and .

2. Some Results of Functions in and

Unless otherwise mentioned, we assume throughout this paper that , , , , and .

Let , we have

Consider

The condition (30) may be rewritten into the form

It follows that the range of the expression , , is a conical domain or where and . Note that is such that and is a curve defined by

Any in is a quadratic equation in two variables and that have no term; it is well known that it is a symmetrical conic section about the real axis (for more details, see [1]). It follows that the domain is bounded by an ellipse for , by a parabola for and by a hyperbola if .

Finally, for , is the right half plane . From (30), we obtain that if and only if, for ,

Making use of the properties of the domain and (36), it follows that if , then

Denote by the class of analytic and normalized Carathéodory functions and by , the function such that . Following the notation applied by Ma and Minda [38], for and , let denote the following class of functions:

The functions which play the role of extremal functions for the class , see [10] (see also [8, 39]) and are defined by with , , where is so such that and is Legendre’s complete elliptic integral of the first kind and the complementary integral of .

Obviously, if , then

For , we have (see [40, 41])

Using the Taylor series in [1, 4], for , we have

Finally, when so that, denoting we get

Let be the extremal function in the class . Then, the relation between the extremal functions in the classes and is given by

Making use of (24), (30), and (46), we obtain the following coefficient relation for :

In particular, by a direct computation, we have

Since and the s are nonnegative, it follows that the s are nonnegative.

Theorem 3. If given by (1) belongs to , then

Proof. Let Using the relation (24) for , we have Since is univalent in , the function is analytic in and . From we have where we used the inequality and equation (48). From this relation (see [4]) and equation (49), we have So, Theorem 3 has been proven.

Theorem 4. If given by (1) belongs to , then

Proof. The result is clearly true for . Let be an integer with , and assume that the inequality is true for all . Making use of (47), we have where we applied the induction hypothesis to and the Rogosinski result (see [42]). Therefore, Applying the principle of mathematical induction, we find that from which the inequality (57) follows.

Similarly, we can prove the following.

Theorem 5. If of the form (1) belongs to the class , then

Theorem 6. If of the form (1) belongs to the class , then

Theorem 7. Let be given by (1). If the inequality holds true, then .

Proof. Making use of the definition (30) it suffices to prove that Observe that The last expression is bounded by if inequality (63) holds.

Similarly, we can prove the following.

Theorem 8. Let be given by (1). If the inequality holds true, then .

Now, we need the following lemmas.

Lemma 9 (see [38]). If is a function with positive real part in and is a complex number, then The result is sharp for the functions given by or .

Lemma 10 (see [38]). If is an analytic function with a positive real part in , then when or , the equality holds if and only if or one of its rotations. If , then the equality holds if and only if or one of its rotations. If , the equality holds if and only if or one of its rotations. If , the equality holds if and only if is the reciprocal of one of the functions such that equality holds in the case of .

Also, the above upper bound is sharp, and it can be improved as follows when :

Theorem 11. If given by (1) belongs to , then

Proof. If , we have where is given by (39). From the definition of subordination, we have where is a Schwarz function with and . Let be a function with positive real part in defined by This gives Using (76) in (73), we obtain For any complex number , we have where Our result now follows by an application of Lemma 9. This completes the proof of Theorem 11.

Example 1. Taking , , and in Theorem 11, we obtain the following result:
If given by (1) satisfies the following inequality then

Similarly, we can prove the following theorem for the subclass

Theorem 12. If given by (1) belongs to , then

Example 2. Taking , , and in Theorem 12, we get the following result:
If given by (1) satisfies the following inequality then

Theorem 13. Let If given by (1) belongs to the class , then Further, if , then If , then

Proof. Applying Lemma 10 to (78) and (79), we can obtain our results asserted by Theorem 13.

Similarly, we can prove the following theorem for the class

Theorem 14. Let If given by (1) belongs to the class , then Further, if , then If , then

Remark 15. Putting in the above results, we obtain the corresponding results for the classes and .

3. Conclusion

By using the concept of the basic (or -) calculus, we have introduced two subclasses and of normalized univalent functions which map the open unit disk onto the generalized conic domain. We have obtained a number of important properties including the coefficient estimates, sufficient conditions, and the Fekete-Szegö inequalities for each of these classes. Our results are connected with those in earlier works which are related to the field of geometric function theory.

Data Availability

No data were used to support this study.

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 the Scientific Research Deanship at University of Ha'il, Saudi Arabia, through project number RG-21050.