Sufficient conditions for certain subclasses of meromorphic p-valent functions

In the present paper, we obtain certain sufficient conditionsfor mero- morphic p-valent functions. Several corollaries and consequences of the main results are also considered.


Let
where U is an open unit disk.A function f (z) in Σ p is said to be meromorphically p-valent starlike of order δ if and only if for some δ( 0 ≤ δ < p).We denote by Σ * p (δ) the class of all meromorphically p-valent starlike of order δ.Further, a function f (z) in Σ p is said to be meromorphically p-valent convex of order δ if and only if  for some δ( 0 ≤ δ < p).We denote by Σ k p (δ) the class of all meromorphically p-valent convex of order δ.A function f (z) belonging to Σ p is said to be meromorphically p-valent close-to-convex of order δ if it satisfies for some δ(0 ≤ δ < p).We denote by Σ c p (δ) the subclass of Σ p consisting of functions which are meromorphically p-valent close-to-convex of order δ in U * .
The object in the present paper is to obtain some sufficient conditions for meromorphic p-valent functions.
In the proofs of our main results, we need the following Jack's Lemma [9]: Lemma 1.1.Let the (non constant) function w(z) be analytic in U with w(0) = 0.

If |w(z)| attains its maximum value on the circle |z|
where m is a real number and m ≥ n where n ≥ 1.

Main Results
With the aid of Lemma 1.1, we derive the next two theorems.
Proof: Let the function w be defined by Then, clearly, w is analytic in U with w(0) = 0. We also find from Suppose there exists a point z 0 ∈ U such that|w(z 0 )| = 1 and |w(z)| < 1, when |z| < |z 0 |.Then by applying Lemma 1.1, there exists m ≥ n such that Then by using (2.4) and (2.5), it follows that which contradicts the given hypothesis.Hence |w(z)| < 1, which implies or equivalently This completes the proof of Theorem 2.1.✷ Theorem 2.2.Let the function f ∈Σ p , satisfies the inequality where (α, β ∈ R, λ ≥ 1, p, n ∈ N) .
Proof: Let the function w be defined by Then by using (3.4) and (3.5), it follows that This evidently completes the proof of Theorem 2.2.✷ Sufficient conditions for certain subclasses of meromorphic... 13

Corollaries and Consequences
In this concluding section, we consider some Corollaries and Consequences of our main results (Theorem 2.1 and Theorem 2.2).