New distortion theorems for functions of positive real part and applications to the partial sums of univalent convex functions

Abstract

New distortion theorems are obtained for the class of functions p ( z ) = 1 + c n z n + ⋯ ( n ≥ 1 ) p(z) = 1 + {c_n}{z^n} + \cdots (n \geq 1) which are analytic and Re p ( z ) > α ( 0 ≤ α > 1 ) \text {Re} p(z) > \alpha (0 \leq \alpha > 1) in the unit disk | z | > 1 |z| > 1 . These are used to obtain new results regarding the partial sums of univalent convex functions.