Differential Sandwich-Type Results for Symmetric Functions Connected with a Q-Analog Integral Operator

In this paper, we obtain some applications of the theory of differential subordination, differential superordination, and sandwich-type results for some subclasses of symmetric functions connected with a q-analog integral operator.


Introduction
The theory of q-analysis has an important role in many areas of mathematics and physics. Jackson [1,2] was the first that gave some application of q-calculus and introduced the q-analog of derivative and integral operator (see also [3]). Let H(U) denote the class of analytic functions in the open unit disk U := {z ∈ C : |z| < 1}, and H[a, m] denote the subclass of functions f ∈ H(U) of the form f (z) = a + a m z m + a m+1 z m+1 + . . . , z ∈ U, with a ∈ C and m ∈ N := {1, 2, . . . }. In addition, let A(m) denote the subclass of functions f ∈ H(U) of the form with m ∈ N, and let A := A(1). We define the integral operator K α n,m : A(m) → A(m), with α > 0 and n ≥ 0, as follows: K α n,m f (z) = z + ∞ ∑ k=m+1 n + 1 n + k α a k z k , z ∈ U.
We note that Definition 1. For f , g ∈ H(U), we say that f is subordinate to g, written f (z) ≺ g(z), if there exists a Schwarz function w, which is analytic in U, with w(0) = 0 and |w(z)| < 1 for all z ∈ U, such that f (z) = g(w(z)), z ∈ U. Furthermore, if the function g is univalent in U, then we have the following equivalence (see [4,5]): Let k, h ∈ H(U), and let ϕ(r, s; z) : C 2 × U → C.
(i) If k satisfies the first order differential subordination then k is said to be a solution of the differential subordination in Equation (5). The function q is called a dominant of the solutions of the differential subordination in Equation (5) if k(z) ≺ q(z) for all the functions k satisfying Equation (5). A dominant q is said to be the best dominant of Equation (5) if q(z) ≺ q(z) for all the dominants q.
(ii) If k satisfies the first order differential superordination then k is called to be a solution of the differential superordination in Equation (6). The function q is called a subordinant of the solutions of the differential superordination in Equation (6) if q(z) ≺ k(z) for all the functions k satisfying Equation (6). A subordinant q is said to be the best subordinant of Equation (6) if q(z) ≺ q(z) for all the subordinants q. Miller and Mocanu [6] obtained conditions on the functions h, q and ϕ for which the following implication holds: Applying these methods, in [7,8], the author studied general classes of first order differential superordinations and superordination-preserving integral operators. Using the results of Bulboacȃ [4] (see also [9,10]), the authors of [11] obtained sufficient conditions for functions f ∈ A to satisfy the double subordination where q 1 and q 2 are univalent functions in U, normalized with q 1 (0) = q 2 (0) = 1. Sakaguchi [12] introduced a class S * s of functions starlike with respect to symmetric points, which consists of functions f ∈ A satisfying the inequality that represents a subclass of close-to-convex functions, and hence univalent in U. Moreover, this class includes the class of convex functions and odd starlike functions with respect to the origin (see [12,13]). In addition, Aouf et al. [14] introduced and studied the class S * s,n T(1, 1) of functions n-starlike with respect to symmetric points, which consists of functions f ∈ A, with a k ≤ 0 for k ≥ 2, and satisfying the inequality where D n is the Sȃlȃgean operator [15]. The classes defined in [12,13] could be generalized by introducing the next class of functions, defined with the aid of the N λ,α n,m,q operator defined as follows: is said to be in the class M λ,α n,m,q (γ, µ, A, B) if it satisfies the subordination condition By specializing the parameters α, λ and q, we obtain the following subclasses: (i) For q → 1 − , we define the class W λ,α n,m (γ, µ, A, B) as follows: where the operator I λ,α n,m is defined by Equation (4); (ii) For q → 1 − , α = 0 and λ = 1, we define the class N γ,µ (m, A, B) that corrects the class defined by Muhammad and Marwan [16] as follows: In this paper, we obtain some sharp differential subordination and superordination results for the functions belonging to the class M λ,α n,m,q (γ, µ, A, B) to try to make a connection between a special subclass of analytic functions whose coefficients are given by the q-analog of integral operator and the differential subordination theory.

Preliminaries
To prove our results, we need the following definition and lemmas.
and the function Ψ is convex, Ψ ∈ H[a, m], and is the best dominant of Equation (9).
and assume that If k is analytic in U and then k(z) ≺ q(z), and q is the best dominant of Equation (10).
and Ψ is convex, Ψ ∈ H [1, m], and is the best dominant.
Proof. If we define the function h by from Equation (7), it follows that h is an analytic function in U, with h(0) = 1. Differentiating Equation (12) with respect to z, we obtain that Moreover, with unknowns γ j , j ≥ 1, we have and equating the corresponding coefficients it follows that According to Equation (12), we have and using the binomial power expansion formula, we get Now, from the subordination in Equation (13), using Lemma 1 for ζ = µ γ , we obtain our result.
Taking q → 1 − in Theorem 1, we obtain the following corollary: for all γ ∈ C with Re γ ≥ 0.
Proof. Since f ∈ A(m) such that Equation (7) holds, it follows that the function h defined by Equation (12) is analytic in U, and h(0) = 1. As in the proof of Theorem 1, differentiating Equation (12) with respect to z, we obtain that Equation (19) is equivalent to Using Lemma 2 for ξ := 1 and ϕ := γ µ , we get that the above subordination implies h(z) ≺ q(z), and q is the best dominant of Equation (19).

we easily check that Equation (22) holds if and only if the assumption in Equation
Taking q → 1 − in Theorem 3, we obtain the following corollary:

Corollary 4.
Suppose that q is univalent in U, with q(0) = 1, and let γ ∈ C * such that If f ∈ A(m) such that Equation (7) holds, and satisfies the subordination and q is the best dominant of Equation (19).

Theorem 4.
Let q be convex in U, with q(0) = 1, and γ ∈ C * , with Re γ ≥ 0. In addition, let f ∈ A(m) such that 2z and assume that the function and q is the best subordinant of Equation (25).

Proof.
Letting the function h be defined by Equation (12), then h ∈ H[q(0), m], and from Equation (23) we have that h ∈ H[q(0), 1] ∩ Q. As in the proof of Theorem 1, differentiating Equation (12) with respect to z, we obtain that Now, according to Lemma 3 for k := γ µ we obtain the desired result.
Taking q(z) = 1 + Az 1 + Bz , with −1 ≤ B < A ≤ 1, in Theorem 4, we obtain the following corollary: 1. If f ∈ A(m) such that the assumptions in Equations (23) and (24) hold, then where Θ and Φ are given in Theorem 5, and e r 1 z and e r 2 z are, respectively, the best subordinant and the best dominant.
2. If f ∈ A(m) such that the assumptions in Equations (23) and (24) hold for the operator N λ,α n,m,q replaced by I λ,α n,m,q , then where Θ and Φ are given in Corollary 7, and e r 1 z and e r 2 z are, respectively, the best subordinant and the best dominant.
A simple computation shows that hence, we obtain and the right-hand side of Equation (31) is positive provided that r < R, where R is given by Equation (29).
Concluding, all the above results give us information about subordination and superordination properties, inclusion results, radius problem, and sharp estimations for the classes M λ,α n,m,q (γ, µ, A, B), together general sharp subordination and superordination for the operator N λ,α n,m,q . For special choices of the parameters γ ∈ C, 0 < µ < 1, −1 ≤ B < A ≤ 1, m ∈ N, α > 0, n ≥ 0, 0 < q < 1, and λ > −1, we may obtain several simple applications connected with the above-mentioned classes and operator.
Author Contributions: The authors contributed equally to this work.