Research Article
Pascu-Type Analytic Functions by Using Mittag-Leffler
Functions in Janowski Domain
Wali Khan Mashwan ,
1
Bakhtiar Ahmad ,
2
Muhammad Ghaffar Khan ,
1
Saima Mustafa,
3
Sama Arjika ,
4,5
and Bilal Khan
6
1
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan
2
Govt: Degree College Mardan, Mardan 23200, Pakistan
3
Department of Mathematics and Statistics, PMAS Arid Agriculture University, Rawalpindi, Pakistan
4
Department of Mathematics and Informatics, University of Agadez, Agadez, Niger
5
International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi,
Post Box 072, Cotonou 50, Benin
6
SchoolofMathematicalSciencesandShanghaiKeyLaboratoryofPMMP,EastChinaNormalUniversity,500DongchuanRoad,
Shanghai 200241, China
Correspondence should be addressed to Sama Arjika; rjksama2008@gmail.com
Received 8 May 2021; Revised 5 July 2021; Accepted 30 July 2021; Published 12 August 2021
Academic Editor: A. M. Bastos Pereira
Copyright © 2021 Wali Khan Mashwan et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Keeping in view the various important applications of Mittag-Leffer functions in the fields of applied sciences, we introduce Pascu-
type analytic functions utilizing the concept of Mittag-Leffler functions in the region of Janowski domain. Moreover, we in-
vestigate some useful properties of these functions such as sufficiency criteria, distortion and growth bounds, convex combination,
radius of starlikeness, and some partial sum results.
1. 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:
f(z)� z +
∞
k�2
a
k
z
k
, (z ∈ D). (1)
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:
R
zf
′
(z)
f(z)
> 0, (z ∈ D). (2)
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 repre-
sented as
h
1
≺ h
2
or h
1
(z) ≺ h
2
(z), (3)
if there exists Schwarz function u, with conditions u(0)� 0
and |u(z)| < 1 such that
h
1
(z)� h
2
(u(z)). (4)
Further if h
2
∈ S, then the above definition is parallel to
h
1
(z) ≺ h
2
(z)(z ∈ D) ⇔ h
1
(0)� h
2
(0),
h
1
(D) ⊂ h
2
(D).
(5)
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 P[A, B] if for
− 1 ≤ B < A ≤ 1,
Hindawi
Mathematical Problems in Engineering
Volume 2021, Article ID 1209871, 7 pages
https://doi.org/10.1155/2021/1209871