On partial sums of certain meromophic -valent functions

In this paper, we study the ratio of meromorphic p -valent functions in the punctured disk U ∗ = { z : 0 < | z | < 1 } of the form f ( z ) = 1 z p + ∑ k = 1 ∞ a k + p − 1 z k + p − 1 to its sequence of partial sums of the form f ( z ) = 1 z p + ∑ k = 1 n + p − 1 a k + p − 1 z k + p − 1 . Also, we determine sharp lower bounds for Re { f ( z ) / f n + p − 1 ( z ) } , Re { f n + p − 1 ( z ) / f ( z ) } , Re { f ′ ( z ) / f n + p − 1 ′ ( z ) } and Re { f n + p − 1 ′ ( z ) / f ′ ( z ) } .