Journal of Mathematics and Statistics 6 (3): 253-260, 2010 ISSN 1549-3644 © 2010 Science Publications 253 On Parametric p-Valent Meromorphic Functions Poonam Sharma Department of Mathematics and Astronomy, University of Lucknow Lucknow, 226007, India Abstract: Problem statement: In this research, we studied parametric p-valent meromorphic functions P α f by considering two classes p M() α β and p M ( ,A) α λ . Approach: With the help of Jack’s Lemma an inclusion relation for the class p M α was obtained and it is shown that this class is closed by an integral operator I c . Results: A subordination result for the class p M ( ,A) α λ 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. Key words: Meromorphic functions, starlike functions, convolution, generalized hypergeometrc functions INTRODUCTION Let M p denotes a class of functions of the form: p k p k p k 1 f (z) z a z ,p N {1, 2,3,...} ∞ - - - = = + ∈ = ∑ (1a) which are analytic and p-valent in the punctured unit disk U * = {z: 0<|z|<1} = U\{0}. We say that a function f(z) ∈ M p is in the class M * p (β) if f(z) ≠ 0 and: zf '(z) Re ,0 p,z U f (z)   < -β ≤β< ∈     Functions in the class M * p (β) are called p-valent meromorphic starlike of order β. Let g α (z) ∈M p be of the form: [ ] ( 29 p k p k p k 1 g (z) z b z ∞ - - α - = = + α ∑ (1b) whose coefficient b k-p ([α]) has a parameter α which is either -p or a positive real and it satisfies the relation: [ ] ( 29 [ ] ( 29 k p k p k b 1 b - - α+   α+ = α   α   (1c) 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: ( 29 [ ] ( 29 ( 29 p k p k p k p k 1 P f(z) f g (z) z a b z g f (z) α ∞ - - α - - = α = * = + α = * ∑ (1d) where, ‘*’ stands for convolution or Hadamard product. We have: [ ] ( 29 1 p k p k p k p k 1 k P f(z) z a b z ∞ α+ - - - - = α+   = + α   α   ∑ 1 z(P f(z))' For p, P f(z) p α α+ α=- ≡- Using (1c) and (1d), we can easily obtain the identity related to parametric p-valent meromorphic functions: 1 z(P f(z))' P f(z) ( p)P f(z) α α+ α =α - α+ (1e) 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;