NEW STRONG DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS

. New strong diﬀerential subordination and superordination results are obtained for meromorphic multivalent quasi-convex functions in the punctured unit disk by investigating appropriate classes of admissible functions. Strong diﬀerential sandwich results are also obtained.


Introduction and Preliminaries
Let Σ p denote the class of all functions f of the form: which are analytic in the punctured unit disk U * = {z ∈ C : 0 < |z| < 1}.
Similarly, f ∈ Σ p is meromorphic multivalent convex if f (z) = 0 and Moreover, a function f ∈ Σ p is called meromorphic multivalent quasi-convex function if there exists a meromorphic multivalent convex function g such that g (z) = 0 and −Re (zf (z)) g (z) > 0 (z ∈ U * ).
Let H(U ) be the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1}. For a positive integer n and a ∈ C, let H [a, n] be the subclass of H(U ) consisting of functions of the form: f (z) = a + a n z n + a n+1 z n+1 + · · · , with H = H [1,1].
Let f and g be members of H(U ). The function f is said to be subordinate to g, or (equivalently) g is said to be superordinate to f , if there exists a Schwarz function w which is analytic in U with w (0) = 0 and |w (z) | < 1(z ∈ U ) such that f (z) = g(w(z)). In such a case, we write f ≺ g or f (z) ≺ g(z), z ∈ U . Furthermore, if the function g is univalent in U , then we have the following equivalent (see [5]) Let G(z, ζ) be analytic in U ×Ū and let f (z) be analytic and univalent in U . Then the function G(z, ζ) is said to be strongly subordinate to f (z) or f (z) is said to be strongly superordinate to G(z, ζ), written as G(z, ζ) ≺≺ f (z), if for ζ ∈Ū = {z ∈ C : |z| ≤ 1}, G(z, ζ) as a function of z is subordinate to f (z). We note that Definition 1.1. [6] Let φ : C 3 × U ×Ū → C and let h be a univalent function in U . If F is analytic in U and satisfies the following (second-order) strong differential subordination: then F is called a solution of the strong differential subordination (1.1). The univalent function q is called a dominant of the solutions of the strong differential subordination or more simply a dominant if F (z) ≺ q(z) for all F satisfying (1.1). A dominantq that satisfiesq(z) ≺ q(z) for all dominants q of (1.1) is said to be the best dominant. Definition 1.2. [7] Let φ : C 3 × U ×Ū → C and let h be analytic function in U . If F and φ (F (z), zF (z), z 2 F (z); z, ζ) are univalent in U for ζ ∈Ū and satisfy the following (second-order) strong differential superordination: then F is called a solution of the strong differential superordination (1.2). An analytic function q is called a subordinant of the solutions of the strong differential superordination or more simply a subordinant if q(z) ≺ F (z) for all F satisfying (1.2). A univalent subordinantq that satisfies q(z) ≺q(z) for all subordinants q of (1.2) is said to be the best subordinant.
and are such that q (ξ) = 0 for ξ ∈ ∂U \E(q). Furthermore, let the subclass of Q for which q(0) = a be denoted by Q(a), Definition 1.4. [9] Let Ω be a set in C, q ∈ Q, and n ∈ N. The class of admissible functions Ψ n [Ω, q] consists of those functions ψ : C 3 × U ×Ū → C that satisfy the following admissibility condition: ψ(r, s, t; z, ζ) / ∈ Ω, whenever Definition 1.5. [8] Let Ω be a set in C and q ∈ H [a, n] with q (z) = 0. The class of admissible functions Ψ n [Ω, q] consists of those functions ψ : C 3 × U ×Ū → C that satisfy the following admissibility condition: ψ(r, s, t; ξ, ζ) ∈ Ω, whenever In our investigations, we will need the following lemmas.
In recent years, several authors obtained many interesting results in strong differential subordination and superordination [1][2][3][4]. In this present investigation, by making use of the strong differential subordination results and strong differential superordination results of Oros and Oros [8,9], we consider certain suitable classes of admissible functions and investigate some strong differential subordination and superordination properties of meromorphic multivalent quasi-convex functions.
Proof. Let the analytic function F in U be defined by After some calculation, we have Further computations show that Define the transforms from C 3 to C by Let (2.5) ψ (r, s, t; z, ζ) = φ (u, v, w; z, ζ) = φ r, s r , r(t + s) − s 2 r 2 ; z, ζ .
To complete the proof, we next show that the admissibility condition for φ ∈ Φ H [Ω, q] is equivalent to the admissibility condition for ψ as given in Definition 1.4. Note that t s We consider the special situation when Ω = C is a simply connected domain. In this case Ω = h(U ), for some conformal mapping h of U onto Ω and the class Φ H [h(U ), q] is written as Φ H [h, q]. The following result is an immediate consequence of Theorem 2.1.

Corollary 2.1. Let β, γ ∈ C and let h be convex in U with h(0) = 1 and
The next result is an extension of Theorem 2.1 to the case where the behavior of q on ∂U is not known.
≺ q ρ (z). The result is now deduced from the fact that q ρ (z) ≺ q(z).

Theorem 2.4.
Let h be univalent in U and φ : C 3 × U ×Ū → C. Suppose that the differential equation has a solution q with q(0) = 1 and satisfies one of the following conditions: and q is the best dominant.
Proof. By applying Theorem 2.2 and Theorem 2.3, we deduce that q is a dominant of (2.7). Since q satisfies (2.8), it is also a solution of (2.7) and therefore q will be dominated by all dominants. Hence, q is the best dominant of (2.7).