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

Saima Noor and Asima Razzaque Basic Science Department, Preparatory Year Deanship, King Faisal University, Al Ahsa, Saudi Arabia Correspondence should be addressed to Asima Razzaque; arazzaque@kfu.edu.sa Received 23 November 2021; Revised 6 January 2022; Accepted 11 March 2022; Published 5 April 2022 Academic Editor: Sarfraz Nawaz Malik Copyright © 2022 Saima Noor and Asima Razzaque. This 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. In this paper, we formulate the q-analogus of differential operator associated with q-Mittag-Leffler function. By using this newly defined operator, we define a new subclass k −USq,γðα, βÞ, 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.


Introduction Definition
Let A denote the class of functions lðzÞ which are analytic in the open unit disk E = fz ∈ ℂ : jzj < 1g, satisfying the condition lð0Þ = 0 and l′ð0Þ = 1 , and for every l ∈ A has the series expansion of the form l z ð Þ = z + 〠 ∞ n=2 a n z n : Let S ⊂ A be the class of all functions which are univalent in E (see [1]). Also, P denotes the well-known Carathéodory class of functions p which are analytic in open unit disk E and has the series expansion of the form For the function l given by (1) and the function g defined by gðzÞ = z + ∑ ∞ n=2 b n z n , the Hadamard product (convolution) l * g of the functions l and g stated by l * g ð Þz = z + 〠 ∞ n=2 a n b n z n : For the analytic functions l,g, l is said to be subordinate to g (indicated as l ≺ gÞ, if there exists a Schwarz function with w 0 ð Þ = 0 and w z ð Þ j j< 1, ð6Þ such that Furthermore, if g is univalent in E, (see [2]); then, we have l z ð Þ ≺ g z ð Þ if and only if l 0 ð Þ = g 0 ð Þ and l E ð Þ ⊂ g E ð Þ, z ∈ E: The class of starlike functions of order αðS * ðαÞÞ in E and the class of convex functions of order αðKðαÞÞ, 0 ≤ α < 1, were defined as follows: It should be noted that where S * and K are the well-known function classes of starlike and convex functions, respectively.
In the year of 1991, Goodman [3] introduced the class UCV 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 k-uniformly convex functions (k -UCV Þ and a related class k − ST was defined by From different viewpoints, the various subclasses of the normalized analytic function of class A have been studied in the field of Geometric Function Theory. To investigate various subclasses of A, many authors have been used the q-calculus as well as the fractional q-calculus. In 1910, Jackson [7] was among the one of few researchers who studied q -calculus operator theory on q-definite integrals and also Trjitzinsky in [8] studied about analytic theory of linear q -difference equations. Curmicheal [9] studied general theory of linear q-difference equations and the first use of q-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 q-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 q-differential operator and defined some new subclasses of analytic functions in open unit disk E. 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 q-differential operator and the generalized symmetric Sàlàgean q-differential operator, respectively. Furthermore, Srivastava et al. [17] defined q-Noor integral operator by using q-calculus operator theory and investigated some subclasses of biunivalent functions in open unit disk.
Here, we give some basic definitions and details of the q -calculus and suppose that 0 < q < 1: For any nonnegative integer n, the q-integer number ½n q is defined by and for any nonnegative integer n, the q-number shift factorial is defined by We note that when q ⟶ 1−, then ½n! = n. The q-difference operator was introduced by Jackson (see in [7]). For l ∈ A, the q-derivative operator or q-difference operator is defined as It is readily deduced from (1) and (14) that We can observe that The familiar Mittag-Leffler function H α ðzÞ introduced by Mittag-Leffler [18] and its generalization H α,β ðzÞ introduced by Wiman [19] which are defined by Recently, Attiya [20] investigated some applications of Mittag-Leffler functions and generalized k-Mittag-Leffler studied by Rehman et al. in [21]. Moreover, Srivastava et al. [22,23] introduced the generalization of Mittag-Leffler functions.

Geometric Interpretation
where The domain Ω k,γ is not always well defined because in general ð1, 0Þ ∉ Ω k,γ (for example, in particular ð1, 0Þ ∉ Ω 2,0:5 ). We see that in [31], the conic domain Ω k ð0, bÞ concides with Ω k,b only when b is chosen according to This means that for Ω k,γ to contain the point ð1, 0Þ,γ must be chosen according as follows: Since p k,γ ðzÞ is convex univalent, the above definition can be written as where 3 Journal of Function Spaces For more detail (see [32,33]).

Proof. Let
where pðzÞ is the analytic in E and pð0Þ = 1: Let pðzÞ = 1 + ∑ ∞ n=1 c n z n and D m q ðα, βÞlðzÞ is given by (21). Then, (49) implies that Now comparing the coefficients of z n , we obtain which implies a n = 1 Using the results that jc n j ≤ jQ 1 j given in ( [33]), we have Let us take δ = jQ 1 j: Then, we have a n j j ≤ δ For n = 2 in (54), we have Hence, for n = 2 the inequality (48) holds. To prove (48), we use mathematical induction, for n = 3 By using (55), we have Hence, (48) holds for n = 3. Now, we suppose that (48) is true for n = t + 1, that is Consider a t+1 j j≤ δ Hence, (48) holds for n = t + 1: Hence, proof is complete.
Proof. Let lðzÞ ∈ k − US m q,γ ðα, βÞ, then there exists a Schwarz function wðzÞ given by (5), such that Let pðzÞ ∈ P be defined as This gives Using (67) in (64) and comparing with (68), we obtain : Journal of Function Spaces For any complex number μ and after some calculation we have where Using a lemma (36) on (70), we have the required result.

Conclusion
In this paper, we formulate the q-analogous of differential operator associated with q-Mittag-Leffler function. By applying newly defined operator, we defined and investigated a new subclass k − US m q,γ ðα, βÞ, 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 q-analogous of differential operator (21) in different classes of analytic functions such as the meromorphic and multivalent functions (see [38][39][40][41][42].

Data Availability
All data are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.