Pascu-Type Analytic Functions by Using Mittag-Leffler Functions in Janowski Domain

Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan Govt: Degree College Mardan, Mardan 23200, Pakistan Department of Mathematics and Statistics, PMAS Arid Agriculture University, Rawalpindi, Pakistan Department of Mathematics and Informatics, University of Agadez, Agadez, Niger International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Post Box 072, Cotonou 50, Benin School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China


Introduction, Definitions, and Motivation
Let A denote the class of functions f that are analytic in the open unit disc D � z ∈ C: |z| < 1 { }, and its Taylor series representation is as follows: Also, the most well-known subclass of A is the class of univalent functions denoted by S.
Further, S * represents the class of starlike functions in D, that is, f ∈ S and maps D on an starlike domain. Mathematically, it will satisfy the following condition: Now let us recall the familiar concept of subordinations; let h 1 and h 2 be two functions of class A, and we call that a function h 1 is subordinated to h 2 and symbolically represented as if there exists Schwarz function u, with conditions u(0) � 0 and |u(z)| < 1 such that Further if h 2 ∈ S, then the above definition is parallel to In 1973, Janowski [1] introduced the concepts of circular domain by introducing Janowski-type functions. A function g (z) analytic in open unit disc with properties that g(0) � 1 is said to be in the class of Janowski denoted by represents an open circular disc centered on real axis with D 1 � 1 − A/1 − B and D 2 � 1 + A/1 + B as diameter end points with 0 < D 1 < 1 < D 2 . After introducing circular domain's concept, several researchers focused on this domain and associated several subclasses to this domain and investigated some of its interesting geometric properties. For some recent works on Janowski-type domain, we refer the readers to [2][3][4][5][6].
e use of special functions in the field of mathematics and applied sciences is very important. One special-type function with single parameter α is defined as and with two parameters α and β, its generalized form is , α, β, z ∈ C and R(α) > 0, R(β) > 0, which are known as Mittag-Leffler functions (see [7]). is was introduced by Mittag-Leffler [8] in connection with his method of summation of some divergent series. Mittag-Leffler in his paper [8] examined and investigated certain very interesting properties of functions defined in form (9).
Mittag-Leffler functions naturally arise in solutions of fractional-order integral equations or fractional-order differential equations. Especially these type of functions can be seen in the investigations of the fractional generalization of the kinetic equation, Levy flights, random walks, superdiffusive transport, and in the study of complex systems. Mittag-Leffler functions, both ordinary and generalized functions, interpolate between a purely exponential law and power-law-like behavior of phenomena governed by ordinary kinetic equations and fractional counterparts (see [9] for more details). During the last 15 years, the interest of Mittag-Leffler functions is considerably increased among engineers and scientists due to their vast applications in several applied problems such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks, probability, and probability distribution (for more applications, see [10,11]). By giving specific values to α and β, we can get the following special cases easily. (1) (2) (3) (4) Now the function is defined by Related to this, Elhaddad et al. [12] defined the operator D m μ (α, β)f(z): A ⟶ A as follows: where m ∈ N 0 � N ∪ 0 { }, μ ≥ 0. Now let us recall some recent Pascu-type works. In 2014, Vijaya et al. [13] defined Pascu-type class for harmonic functions with positive coefficient involving Salagean operator. After that, in 2015, Murugusundaramoorty et al. [14] found coefficient estimates for Pascu-type subclass of bi-univalent functions based on subordination. Recently, Vijaya [15] obtained coefficient bounds for subclass of Pascu-type bi-starlike functions in parabolic domain.
where the notation "≺" stands for the familiar concept of subordinations.
It can easily be easily verified that a function f ∈ A will be in the class S m,t μ (α, β, A, B), if and only if

The Main Results and Their Consequences
In this section, we start with sufficiency criteria for this newly defined class. β, A, B), it will be enough if we show that inequality (18) holds true, i.e., where Now consider Here we have used inequality (19), and this completes the direct part of the proof.

Mathematical Problems in Engineering
Now we choose values of z on the real axis so that (1 − t) zD m+1 is real. Upon clearing the denominator in (2) and letting z ⟶ 1 − through real values, we obtain (19).
where β, A, B), and the result is sharp for this function.

Corollary 1. Let f ∈ A of form (1) be in the class
where Λ k and σ k are given above for k ≥ 2.
In the following, we discuss the growth and distortion theorems for our new class of functions.
Theorem 2. If f is in the class S m,t μ (α, β, A, B) and has form (1). en, for |z| � r, we have ese inequalities are sharp for the function defined as Proof. First, we want to prove inequality (28). For this, consider since for |z| � r < 1 we have r n ≤ r 2 for n ≥ 2 and Similarly, But Hence, Now by putting this value in (32) and (33), we get the required result. e proof of (29) is similar to (28), so it is omitted. Proof. To show the class S m,t μ (α, β, A, B) is closed under convex combination, it is enough to show that β, A, B), where f 1 and f 2 ∈ S m,t μ (α, β, A, B) and μ ∈ [0, 1]. Since by eorem 1, we have Hence, in light of eorem 1, it is the required result. β, A, B) if and only if
Proof. Let (41) hold. en, Hence, by β, A, B). β, A, B); then, we have and c 1 � 1 − ∞ k�2 c k . en, function is of the form given by (40), and thus this completes the proof. β, A, B); then, f is starlike function of order δ in |z| < r * , where . (44) Proof. We know that f is starlike function of order δ if and only if Using (8) and upon simplification yields In light of eorem 1, we can easily obtain which implies that . (49) Hence, we obtain the desired result. □ Theorem 6. If f is in the class S m,t μ (α, β, A, B) and has form (1). en, where Proof. First we prove (50); for this, let us write where if and only if m k�2 a k + c k+1 ∞ k�m+1 a k ≤ 1.
In order to prove the inequality in (50), it is enough to show that the left-hand side of (55) is bounded above by the following sum: and is equivalent to which is true for inequality (50). Next to derive inequality (51), let us write where Mathematical Problems in Engineering is equivalent to the following inequality: Finally, we can see that the left-hand side of this inequality is bounded above by the following sum: which is equivalent to where c k � Λ k σ k |B − A| , c k ≥ 1 for k � 2, 3, . . . , m c k+1 for k � m + 1, . . . .
Proof. e proof is quite the same as eorem 6, so it is omitted.

Conclusion
e present article is the motivation of some applications of Mittag-Leffer functions in various fields of applied sciences. A subclass of analytic functions is introduced using the concepts of Pascu-type and Mittag-Leffler functions in the region of Janowski domain. Some investigations are discussed with aid of some useful properties such as sufficiency criteria, distortion and growth bounds, convex combination, radius of starlikeness, and some partial sum results.
is idea can be extended to various other classes like meromorphic functions, biunivalent functions, and harmonic functions.

Data Availability
No data were used to support this study.