An Application of Salagean Derivative on Partial Sums of Certain Analytic and Univalent Functions

Let φ (z) be a fixed analytic and univalent function of the form φ(z) = z + ∑∞ k=2 ckz k and Hφ (ck, δ) be the subclass consisting of analytic and univalent functions f of the form f(z) = z + ∑∞ k=2 akz k which satisfy the inequality ∑∞ k=2 ck |ak| ≤ δ. In this paper, we determine the sharp lower bounds for Re { Dpf(z) Dfn(z) } and Re { Dfn(z) Dpf(z) } , where fn(z) = z + ∑n k=2 akz k be the sequence of partial sums of a function f(z) = z+ ∑∞ k=2 akz k belonging to the class Hφ (ck, δ) and Dp stands for the Salagean derivative. In this paper, we extend the results of ([1], [2], [3], [6]) and we point out that some conditions on the results of Frasin (([1],Theorem 2), ([2],Theorem 2.7)) are incorrect and we correct them. 2010 Mathematics Subject Classification: 30C45.