On Parametric p-Valent Meromorphic Functions

Problem statement: In this research, we studied parametric p-valent meromorphic functions Pαf by considering two classes Mαp (β) and Mαp (λ,A) . Approach: With the help of Jack’s Lemma an inclusion relation for the class Mαp was obtained and it is shown that this class is closed by an integral operator Ic. Results: A subordination result for the class Mαp (λ,A) was proved. Consequences of main results with the results for special values of the parameter α were discussed. Conclusion/Recommendations: Our results certainly generalized several results obtained earlier as well as generate new results.


INTRODUCTION
Let M p denotes a class of functions of the form: For g α (z) given by (1b), a parametric convolution operator P α : M p →M p , on the function f(z) of the form (1a) is defined by: where, '*' stands for convolution or Hadamard product. We have: Using (1c) and (1d), we can easily obtain the identity related to parametric p-valent meromorphic functions: Several subclasses of p-valent meromorphic functions involving various convolution operators have been defined and studied in (Aouf, 2008;Liu and Srivastava, 2001;2004;Raina and Srivastava, 2006;Srivastava and Patel, 2006;Srivastava et al., 2008;Wang et al., 2009;Yang, 2001). The purpose of this study is to unify the results obtained earlier and to give some new results. Motivated with these earlier works especially the work of Cho (Srivastava and Owa, 1992;Yang, 2004), in this study, in order to study parametric p-valent meromorphic functions P f (z) satisfying following conditions respectively: and for λ>p,|A|<|α|: where, '≺' stands for subordination between two analytic functions in U.
Theorem 2: Let an integral operator I c : M p →M p be defined for c>0 by: The class p M α , defined in (1f) is closed under the integral operator I c .

Remark:
The results obtained in Theorems 1 and 2, coincide for α = n + p>0, with the results obtained by Cho in (Srivastava and Owa, 1992).

Proof:
Let which is univalent, convex in U. Consider for λ>p: ,(w C) λ θ = ϕ = ∈ α which are analytic in C so that: We obtain that: which is starlike in U and: Hence on applying Lemma 1, we get that p(z)≺q(z) or αz p P α f(z)≺α+Az, α+Az is the best dominant and the result is sharp with extremal function:

RESULTS AND DISCUSSION
Some consequences of the results are discussed along with some special cases as follows: For positive real numbers a i , b i (i = 1,2,…,q) and for positive integers A i , B i (i = 1,2,…,q) such that We get for f(z)∈M p and for α>0: where, q+1 ψ q (z) is Wright's (psi) function which is a generalized hypergeometric function (Srivastava and Manocha, 1984) and its series representation is given by:   Aouf and Dziok, 2008a;2008b;Sharma, 2010;. Taking A i = B i = 1, i = 1,2,…,q, q+1 ψ q (z) reduces to the generalized hypergeometric function q +1 F q and we denote: The operator similar to F([a 1 ])f, has been studied recently by Wang et al. (2009); Raina and Srivastava (2006); Aouf (2008) and Liu and Srivastava (2004). Taking q = 2 and b 2 = 1, we get the operator, involving Gauss's hypergeometric function 2 F 1 : Further, taking q = 1, we get: which is studied extensively by Liu and Srivastava (2001); Liu (2000); Aouf and Srivastava (2006) and Srivastava et al. (2008). If a 1 = n + p, n∈N and b 1 = 1, we get, Ruscheweyh derivative operator for M p class: which is studied by Cho in (Srivastava and Owa, 1992;Uralegaddi and Somanatha, 1992;Joshi and Srivastava, 1999;Aouf, 1993) and its special case is studied by Aouf and Hossen (1994).
From Theorem 1, we directly get following results.