On Quantum Differential Subordination Related with Certain Family of Analytic Functions

Recently, there is a rapid increase of research in the area of Quantum calculus (known as q-calculus) due to its widespread applications in many areas of study, such as geometric functions theory. To this end, using the concept of q-conic domains of Janowski type as well as qcalculus, new subclasses of analytic functions are introduced. (is family of functions extends the notion of α-convex and quasi-convex functions. Furthermore, a coefficient inequality, sufficiency criteria, and covering results for these novel classes are derived. Besides, some remarkable consequences of our investigation are highlighted.


Introduction
Recently, there is a rapid increase in the area of Quantum calculus (known as q-calculus) due to its widespread applications in many areas of study such as geometry functions theory (GFT), combinatorial analysis, Lie theory, mechanical engineering, cosmology, and statistics. e concept of q-integral was first introduced and studied by Jackson et al. [6] at the beginning of the twentieth century. e development of the concept of q-calculus in GFT had its history from the work of Ismail et al. [5], where the notion of q-starlike functions was extensively studied. As such, many subclasses of univalent functions correlated with q-calculus have been on increase (see [1, 14, 17, 20, 23-25, 27, 28]). In recent times, various family of qextension of starlike functions, which are connected to Janowski functions in the open unit disc U, were initiated and examined from many different viewpoints and perspective (see [17,20,28]).
In an attempt to generalize the notion of uniformly closed-to-convex functions considered by Goodman [3], Kanas and Wisniowska [8][9][10]13] and Kanas and Srivastava [12] introduced the conic domain Ω m (m ≥ 0) and studied the classes m − UCV and m − UST of m-uniformly convex and starlike functions. Furthermore, Noor and Malik [22], using the concept of Janowski class, extended the domain Ω m to Ω m (c, λ), − 1 ≤ λ < c ≤ 1. In the latest article by Mahmood et al. [17], the importance of q-calculus was used to improve the Noor-Malik conic domains to Ω q,m (c, λ). Using this domain, they examined the coefficient inequalities associated with the class m − UST q (c, λ) of q-uniformly starlike functions. Afterward, the same coefficient problems were also explored for the classes of m-uniformly q-convex, close-to-convex and quasi-convex functions by Naeem et al. [20].
Motivated by these recent articles [15,17,20,28], our aim is to introduce the novel classes m − UM q (α, c, λ) and m − UQ q (α, c, λ) consisting of m-uniformly q-alpha convex and quasi-convex functions. We study the coefficient inequalities associated with these classes and some other related properties. Some relevant consequences of our results which were studied in previous work show the significance of our investigation.

Materials and Methods
Now, we give some useful preliminaries which are necessary for our study.
Let A be the class of normalized analytic functions f(ς) Let S, CV, ST, QV, and KV be the subclasses of A consisting functions that are univalent, convex, starlike, quasi-convex, and close-to-convex functions, respectively. e function f(ς) of form (1) is subordinate to the analytic function g(ς) (written as f(ς)≺g(ς)) of the form if there exists a Schwarz function w(ς) in U such that Let − 1 ≤ λ < c ≤ 1. en, class P(c, λ) (see [7]) of function p(ς) satisfies the subordination condition or equivalently, where h ∈ P (class of functions with positive real part). For c � 1 − 2β, 0 ≤ β < 1, and λ � − 1, the class P(c, λ) reduces to the class P(β), the class of functions whose real part is greater than β. e conic domains Ω m (c, λ)(m ≥ 0) of Janowski type introduced by Noor and Malik [22] are defined as follows: Geometrical interpretation of Ω m (c, λ) and its effect on Ω m were also demonstrated in [22]. e class m − P(c, λ) represents the class of all functions that maps U onto Ω m (c, λ). Equivalently, a function p(ς) belongs to m − P(c, λ) if and only if where p m (ς) has its definition in [10,11] and given by , t ∈ (0, 1), ς ∈ U and t is chosen such that m � cosh((πR ′ (t))/(4R(t))), R(t) is Legendre's complete elliptic integral of the first kind, and R ′ (t) is the complementary integral of R(t) Definition 1 (see [2]). Let q ∈ (0, 1). en, the q-number [n] q is given as q ι � 1 + q + q 2 + · · · + q n− 1 , n ∈ N, n, as q ⟶ 1 − , and the q-derivative of a complex valued function f(ς) in U is given by From the above explanation, it is easy to see that, for f(ς) given by (1), Definition 2 (see [28]). An analytic function p(ς) in U belongs to P q (c, λ) if and only if the condition Definition 3 (see [17]). An analytic function p(ς) in U belongs to m − P q (c, λ) if and only if the subordination condition is satisfied, where p m (ς) is given by (8). Equivalently, where Equivalently, We note that Definition 4 (see [20]). Let g(ς) of form (2) be in A and where p m (ς) is given by (8). Inspired by the above recent mentioned work, we announce the following novel classes of analytic functions.
To effectively establish our findings, the following set of lemmas is required.
Lemma 2 (see [17]). If p(ς) � 1 + c 1 ς + c 2 ς 2 + · · · ∈ P, then, for any real number δ, or one of its rotations. If δ � 1, then the equality holds if and only if p(ς) is the reciprocal of one of the functions such that equality holds for the case δ � 0. Although the above upper bound is sharp, when 0 < δ < 1 it can be improved as follows: Lemma 3 (see [20]). Let g(ς) be of form (2) and where Q 1 is defined by (23).

Results and Discussion
We now turn our attention to the main results of this article.

Theorem 1. A function f(ς) of form (1) belongs to the class
where Proof. Suppose condition (29) holds. en, we need to prove that erefore,

Journal of Mathematics 5
We have [n] q a n ς n ⎛ ⎝ ⎞ ⎠ ∞ n�0 [n] q a n ς n− 1 Similarly, [n] 2 q + 1 a n + From inequality (32) and equations (33) and (34), we arrive at Journal of Mathematics e last inequality is bounded by 1 if (29) is satisfied. is completes the proof.

Corollary 1. A function f(ς) of type (1) is in
Setting α � 0 in eorem 1, we obtain a variance version of the result presented in [17].

Journal of Mathematics
Following the same process in the proof of eorem 1, we have where we have used Lemma 3. us, the last inequality is bounded by 1 if (42) is satisfied. Hence, we complete the proof.
As q ⟶ 1 − in eorem 2, we obtain the similar result proved in [18].