STUDIA UNIV. “BABES ¸–BOLYAI”, MATHEMATICA, Volume L, Number 4, December 2005 PARTIAL SUMS OF CERTAIN MEROMORPHIC P-VALENT FUNCTIONS B.A. FRASIN AND G. MURUGUSUNDARAMOORTHY Abstract. In this paper, we study the ratio of meromorphic p-valent functions in the punctured disk D = {z :0 < |z| < 1} of the form f (z)= 1 z p + ∞ ∑ k=1 a p+k−1 z p+k−1 to its sequence of partial sums of the form fn(z) = 1 z p + n ∑ k=1 a p+k−1 z p+k−1 . Also, we will determine sharp lower bounds for Re {f (z)/fn(z)} , Re {fn(z)/f (z)} , Re {f ′ (z)/f ′ n (z)} and Re  f ′ n (z)/f ′ (z)  . 1. Introduction and definitions Let Σ p denotes the class of functions of the form: f (z)= 1 z p + ∞  k=1 a p+k−1 z p+k−1 (p ∈ N), (1) which are analytic and p-valent in the punctured unit disk D = {z :0 < |z| < 1}. A function f ∈ Σ p is said to be in the class Σ ∗ (p, α) of meromorphic p-valently starlike functions of order α in D if and only if Re  - zf ′ (z) f (z)  >α (z ∈D;0 ≤ α<p; p ∈ N). (2) Furthermore, a function f ∈ Σ p is said to be in the class Σ K (p, α) of meromorphic p-valently convex functions of order α in D if and only if Re  -1 - zf ′′ (z) f ′ (z)  >α (z ∈D;0 ≤ α<p; p ∈ N). (3) Received by the editors: 10.05.2005. 2000 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. Meromorphic p-valent functions, meromorphic p-valently starlike and meromorphic p-valently convex functions, partial sums. 33