On a New Criterion for Meromorphic Starlike Functions

and Applied Analysis 3 we have E = 󵄨 󵄨 󵄨 󵄨 q (z 0 ) [(1 − α) βi + α − λ] + λ − 1 󵄨 󵄨 󵄨 󵄨 2 = (u 2 + V2) [(1 − α)2β2 + (α − λ)2] − 2 (1 − λ)R ((u + iV) [(1 − α) βi + α − λ]) + (1 − λ)2 = (u 2 + V2) (1 − α)2β2 + 2 (1 − λ) (1 − α) βV + |(u + iV) (α − λ) − (1 − λ)|2. (31) By means of (24), we obtain |(u + iV) (α − λ) − (1 − λ)| = |(u + iV) (α − λ) − (α − λ) + α − λ − 1 + λ| = |(α − λ) (u + iV − 1) − (1 − α)| ≧ 1 − α − (λ − α) |u + iV − 1| ≧ 1 − α − (λ − α)N. (32) It follows from (31) and (32) that E ≧ (u 2 + V2) (1 − α)2β2 + 2 (1 − λ) (1 − α) βV + [1 − α − (λ − α)N] 2 . (33) We now set


Introduction and Preliminaries
A function  ∈ Σ  is said to be in the class MS *  () of meromorphic starlike functions of order  if it satisfies the condition R (   ()  () ) < − ( ∈ U; 0 ≦  < 1) .
For simplicity, we write MS *  (0) =: MS *  .For two functions  and , analytic in U, we say that the function  is subordinate to  in U and write if there exists a Schwarz function , which is analytic in U with such that Furthermore, if the function  is univalent in U, then we have the following equivalence: In a recent paper, Miller et al. [1] proved the following result.
More recently, Catas [2] improved Theorem A as follows.
In order to prove our main results, we require the following subordination result due to Hallenbeck and Ruscheweyh [13].
Lemma 1.Let  be a convex function with (0) = 1, and let  ̸ = 0 be a complex number with R() ≧ 0. If a function satisfies the condition (16)

Main Results
We begin by stating the following result.
Taking  = 0 in Theorem 2, we obtain the following result.