Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points

In the present research paper, our aim is to introduce a new subfamily of meromorphic p-valent (multivalent) functions. Moreover, we investigate sufficiency criterion for such defined family.


Introduction
Let the notation Ω p be the family of meromorphic p-valent functions f that are holomorphic (analytic) in the region of punctured disk E = {z ∈ C : 0 < |z| < 1} and obeying the following normalization In particular Ω 1 = Ω, the familiar set of meromorphic functions.Further, the symbol MS * represents the set of meromorphic starlike functions which is a subfamily of Ω and is given by Two points p and p are said to be symmetrical with respect to o if o is the midpoint of the line segment pp .This idea was further nourished in [1,2] by introducing the family MS * s which is defined in set builder form as; 2 and p ∈ N, we introduce a subfamily of Ω p consisting of all meromorphic p-valent functions of reciprocal order ξ, denoted by N S λ p (s, t, ξ), and is defined by We note that for p = s = 1 and t = −1, the class N S λ p (s, t, ξ) reduces to the class N S λ 1 (1, −1, ξ) = N S λ * (ξ) and is represented by For detail of the related topics, see the work of Al-Amiri and Mocanu [3], Rosihan and Ravichandran [4], Aouf and Hossen [5], Arif [6], Goyal and Prajapat [7], Joshi and Srivastava [8], Liu and Srivastava [9], Raina and Srivastava [10], Sun et al. [11], Shi et al. [12] and Owa et al. [13], see also [14][15][16].
For simplicity and ignoring the repetition, we state here the constraints on each parameter as 0 < ξ < 1, −1 ≤ t < s ≤ 1 with s = 0 = t, λ is real with |λ| < π 2 and p ∈ N. We need to mention the following lemmas which will use in the main results.
This result is the best possible." 2. Sufficiency Criterion for the Family N S λ p (s, t, ξ) In this section, we investigate the sufficiency criterion for any meromorphic p-valent functions belonging to the introduced family N S λ p (s, t, ξ) : Now, we obtain the necessary and sufficient condition for the p-valent function f to be in the family N S λ p (s, t, ξ) as follows: Theorem 1.Let the function f (z) be the member of the family Ω p .Then where Proof.Suppose that inequality (2) holds.Then, we have and hence the result follows.
Next, we investigate the sufficient condition for the p-valent function f to be in the family N S λ p (s, t, ξ) in the following theorem: Theorem 2. If f (z) belongs to the family Ω p of meromorphic p-valent functions and obeying Proof.To prove the required result we only need to show that 2e iλ ξ cos λz f (z) /p − s p t p (t p −s p ) ( f (tz) − f (sz)) Now consider the left hand side of (5), we get . By virtue of inequality (4), we at once get the desired result.
Also, we obtain another sufficient condition for the p-valent function f to be in the family N S λ p (s, t, ξ) by using Lemma 1, in the following theorem: , where G (z) is defined in Equation (3).
Proof.Let we choose the function q (z) by then Equation (6) shows that q (z) is holomorphic in E and also normalized by q (0) = 1.
From Equation (6) , we can easily obtain that Now, let us define a set as then, we see that Φ (ia, b, z) / ∈ H and Φ (q (z) , zq (z) , z) ∈ H. Therefore, by using Lemma 1, we obtain that (q (z)) > 0.
Further, in the next theorem, we obtain the sufficient condition for the p-valent function f to be in the family N S λ p (s, t, ξ) by using Lemma 2.
Theorem 4. If f (z) is a member of the family Ω p of meromorphic p-valent functions and satisfies e iλ G (z) then f (z) ∈ N S λ p (s, t, ξ) , where G (z) is given by Equation (3).
Proof.In order to prove the required result, we need to define the following function q (z) cos λ = e −iλ G (z) + i sin λ, then, Equation (6) shows that th function q (z) is holomorphic in E and also normalized by q (0) = 1.Now, by routine computations, we get Now, let us consider z 1 q(z) cos λ−i sin λ and then by using inequality (7), we have Using Lemma 2, we have After simplifications, we get The region H (E) shows that it is symmetric about the real axis and also H (z) is convex.Hence (G (z)) ≥ H (1) > 0, or (q (z) cos λ − i sin λ) > β cos λ, or e −iλ G (z) > β cos λ, f or Finally, we investigate the sufficient condition for the p-valent function f to be in the family N S λ p (s, t, ξ) in the following theorem: then f (z) ∈ N S λ p (s, t, ξ) , where G (z) is defined in Equation (3).
Proof.Let us put