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Some Inclusion Results of Operators Associated with a Generalization of the Mittag-Leffler Function
- Jamal Salah
- 76-83
- Sep 10, 2023
- Mathematics
Some Inclusion Results of Operators Associated with a Generalization of the Mittag-Leffler Function
Jamal Salah
Department of Basic Science, College of Applied and Health Science, A’Sharqiyah University, Post Code 42, 400 Ibra, Oman
DOI: https://doi.org/10.51584/IJRIAS.2023.8809
Received: 14 July 2023; Revised: 10 August 2023; Accepted: 14 August 2023; Published: 10 September 2023
ABSTRACT
In this study, we consider one of the generalizations of the well-known Mittag-Leffler function, namely Eθα,β(z). We normalize the latter by multiplication with the factor zΓ(β) to generate a power series that belongs to the well-known class of analytic functions A , in the unit disk D. Consequently, and using spiral-like functions, we investigate some inclusion results.
Keywords: Convolution, Inclusion results, Integral Transform, Spiral-like functions, Mittag-Leffler function Mathematics Subject Classification: 30C45, 30C50
PRELIMINARIES
One of the functions that characterize exponential behavior was developed by a Swedish mathematician and is known as “The Mittag-Leffler Function” (see [1]). The Mittag-Leffler function (M-L) has become more significant due to its widespread applicability in many scientific and technical domains. In some branches of the physical and applied sciences, including probability and statistical distribution theory, fluid mechanics, biological issues, electrical networks, and others, the (M-L) function has been used recently. Lev́y flights, random walks, and—most importantly—generalization of kinetic equations are examples of integro-differential equations in which this function naturally occurs [2, 3]. The (M-L) function has been studied extensively in the literature for its normalization, generalization, characteristics, applications, and extension. One can check out [4, 5] and [6] for further information. The study of fractional generalization of kinetic equations, random walks, Lev́y flights, super-diffusive transport, complex systems, and delayed fractional reaction diffusion all involve fractional-order differential and integral equations; the solutions invariably contain (M-L) function (see [7–10]). Recently, the one-parameter (M-L) function has also been suggested as a solution for mathematical models in biology and tourism (see [11,12]).
Initially, the one-parameter (M-L) function E_α (z) for α∈C, with Re (α>0) (see [13] and [14]) is defined as:
Let S be the subclass of A whose members are univalent in 
. Robertson [22] studied two well-known subclasses of S, namely, the classes of starlike and convex functions. Function f ∈ A given by (1.1) is said to be starlike of order γ,0≤γ<1, if and only if 
and the function class is denoted as S*(γ). We also write S*(0)= : S*, where S* denotes the class of functions f ∈ A such that f 
is starlike domain with respect to the origin. Function f ∈ A is said to be convex of order γ,0≤γ<1, if and only if 
and the class is denoted as K(γ). Furthermore, K:=K(0) represents the well-known standard class of convex functions. By Alexander’s duality relation (see [23]), we know that 
. Function f ∈ A is said to be spiral-like if
. Robertson [22] studied two well-known subclasses of S, namely, the classes of starlike and convex functions. Function f ∈ A given by (1.1) is said to be starlike of order γ,0≤γ<1, if and only if 
and the function class is denoted as S*(γ). We also write S*(0)= : S*, where S* denotes the class of functions f ∈ A such that f is starlike domain with respect to the origin. Function f ∈ A is said to be convex of order γ,0≤γ<1, if and only if 
and the class is denoted as K(γ). Furthermore, K:=K(0) represents the well-known standard class of convex functions. By Alexander’s duality relation (see [23]), we know that 
. Function f ∈ A is said to be spiral-like if
INCLUSION RESULTS FOR THE NORMALIZED (M-L)
We follow the same approach of authors in [28-30], who studied the two-parameter (M-L): E(α,β)(z). Here, we extend the results by involving the three-parameter (M-L) Eθα,β. Prior to proving our main results, we compute the following:
then 
where Eθα,β is defined by (1.2)
INCLUSION RESULTS FOR THE IMAGE OF A LINEAR OPERATOR
First, we introduce the following linear operator Λβα:A→A by the means of Hadamard product
Next, we explore the sufficient conditions for the images of the linear operator Λβα on functions of the class Rτ (ϑ,δ). Thus, we provide sufficient conditions such that these images are in the classes S(ξ,γ,ρ) and K(ξ,γ,ρ), respectively.
Theorem 3.1. For f(z) ∈ A. If
Using Lemma 1.2 and following the same procedure as in the proof of Theorem 2.2, we obtain the following result:
We deduce that
INCLUSION RESULTS FOR THE ALEXANDER INTEGRAL OPERATOR
Theorem 4.1. For the Alexander Integral Operator Ψβα be given by
The proof of Theorem is parallel to that of Theorem 2.1.
CONCLUSIONS
In this study, we normalized the generalized (M-L) to deduce the analytic form E θ (α,β), that we investigated its inclusion results in the subclasses the classes S(ξ,γ,ρ) and K(ξ,γ,ρ). In addition, we discuss sufficient conditions for the linear operator Λβα (f), f ∈ Rτ(ϑ,δ) to be a member of the same subclasses, i.e. S(ξ,γ,ρ) and K(ξ,γ,ρ). Finally, the investigation has been extended to involve Alexander operator Ψβα by the means of Hadamard product.
Conflict of Interest
The author has no conflicts of interest to declare.
FUNDING STATEMENT
This research received no specific grant from any funding agency.
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