Sandwich type results for meromorphic functions with respect to symmetrical points

: In the present paper, we use the technique of differential subordination and superordination involving meromorphic functions with respect to symmetric points and also derive some sandwich results. As a consequence of main result, we obtain results for meromorphic starlike functions with respect to symmetrical points.

The class of such functions is denoted by MS * (α) and write MS * = MS * (0)-the class of meromorphic starlike functions.
In 1959, Sakaguchi [1] introduced and studied the class of starlike functions with respect to symmetric points in E. Further investigations into the class of starlike functions with respect to symmetric points can be found in [2,3].
Recently, Ghaffar et al., [4] introduced and investigated a class of meromorphic starlike functions with respect to symmetric points which satisfies the condition and −1 ≤ B < A ≤ 1. We denote the above class by MS s * [A, B]. If f is analytic and g is analytic univalent in open unit disk E, we say that f (z) is subordinate to g(z) in E and written as f (z) ≺ g(z) if f (0) = g(0) and f (E) ⊂ g(E). To derive certain sandwich-type results, we use the dual concept of differential subordination and superordination.
Let Φ : C 2 × E −→ C (C is the complex plane) and h be univalent in E. If p is analytic in E and satisfies the differential subordination then p is called a solution of the differential subordination (1). The univalent function q is called a dominant of differential subordination (1) if p ≺ q for all p satisfying (1). A dominantq ≺ q for all dominants q of (1), is said to be the best dominant of (1).
Let Ψ : C 2 × E −→ C (C is the complex plane) be analytic and univalent in domain C 2 × E, h be analytic in E, p is analytic and univalent in E, with (p(z), zp (z); z) ∈ C 2 × E for all z ∈ E. Then p is called a solution of first order differential superordination if it satisfies An analytic function q is called a subordinant of differential superordination (2) if q ≺ p for all p satisfying (2). A univalent subordinantq that satisfies q ≺q for all subordinants q for (2), is said to be the best subordinant of (2).
In this paper we study the concepts of subordination and superordination to obtain meromorphic starlikeness with respect to symmetric points.On the basis of the theory we also investigate some important sandwich results of symmtric meromorphic functions.

Preliminaries
We shall use the following lemmas to prove our result. Lemma 1. [5] Let q be univalent in E and let θ and φ be analytic in a domain D containing q(E), with φ(w) = 0, when In addition, assume that If p is analytic in E, with p(0) = q(0), p(E) ⊂ D and then p(z) ≺ q(z) and q is the best dominant.

Definition 1.
We denote by Q the set of functions p that are analytic and injective on E \ B(p), where and are such that p (ζ) = 0 for ζ ∈ ∂E \ B(p).

Lemma 2. [6]
Let q be the univalent in E and let θ and φ be analytic in a domain D containing q(E).
then q(z) ≺ p(z) and q is the best subordinant.

Subordination results
Theorem 1. Let q be univalent in E, with q(0) = 1, and let where γ ∈ C * := C \ {0}. If f ∈ Σ satisfy the condition and and q is the best dominant of (5).
If we take q(z) It is easy to see that the function is a convex domain symmetric with respect to the real axis. Hence Thus, the inequality (6) is equivalent to 1 λ ≥ |B| − 1 |B| + 1 , hence we deduce the following corollary: (4) and . Moreover, the function 1 + Az 1 + Bz is the best dominant of (8).
For A = 1 and B = −1, the above corollary reduces to the next special case: (4) and i.e., f ∈ MS s * , or f is meromorphic starlike with respect to symmetrical points in E. Moreover, the function 1 + z 1 − z is the best dominant of (9).

Theorem 2.
Suppose that q be univalent in E with q(0) = 1 and q(z) = 0 for all z ∈ E such that where γ, µ ∈ C * and ν, η ∈ C with ν + η = 0 and let f ∈ Σ satisfy the conditions If and q is the best dominant of (13) (the power is the principal one).

Superordination and sandwich theorems
Theorem 4. Let q be convex in E with q(0) = 1 and λ ∈ C with λ > 0. Let f ∈ Σ satisfy the condition (4) such ∈ Q and suppose that the function and q is the best subordinant of (26).

Proof. Setting
then p is analytic in E with p(0) = 1. Taking logarithmic differentiation of the above relation with respect to z, we have and a simple calculation yields that the assumption (26) is equivalent to q(z) + λzq (z) ≺ p(z) + λzp (z). Now, in order to prove our result we will use Lemma 2. Consider the functions θ(w) = w and φ(w) = λ analytic in C and set h(z) = zq (z)φ(q(z)) = λzq (z).
Since h(0) = 0, h (0) = λq (0) = 0 and q is convex in E, it follows that h is starlike in E and Therefore, Lemma 2 and assumption (26) imply q(z) ≺ p(z) and the function q is the best subordinant of (26).
Taking q(z) = 1 + Az 1 + Bz in Theorem 4, where −1 ≤ B < A ≤ 1, we get the following corollary: Corollary 8. Let q be convex in E with q(0) = 1 and λ ∈ C with λ > 0. Let f ∈ Σ satisfy the condition (4) such ∈ Q and suppose that the function is univalent in E. If and q is the best subordinant of (27).
Using the same techniques as in proof of Theorem 3 and then applying Lemma 2, we could prove the next theorem: Theorem 5. Let γ, µ ∈ C * and δ, ν, η ∈ C with ν + η = 0 and δ γ > 0. Suppose that q is convex in E, with q(0) = 1 and let f ∈ Σ satisfy the conditions (11), (12) and If the function φ given by (21) is univalent in E and and q is the best subordinant of (28) (the power is the principal one).
If we combine Theorem 1 with Theorem 4 and Theorem 3 with Theorem 5, we deduce the following sandwich results, respectively: Theorem 6. Let q 1 and q 2 be two convex functions in E with q 1 (0) = q 2 (0) = 1 and λ ∈ C with λ > 0. Let f ∈ Σ satisfy the condition (4), such that − 2z f (z) f (z) − f (−z) ∈ Q and suppose that the function then where q 1 and q 2 are respectively the best subordinant and the best dominant of (29).