Some Geometric Properties of Close- to- Convex Functions

Abstract

The objective of this paper is to present some geometric properties of the close to convex function  f when f is an analytic, univalent self-conformal mapping defined in the open unit disk D={z∈C: |z|<1},  and on the boundary of D. One of the goals of this work was determining the sharp bound for the function of form  f(z)=z/(1-z)^2δ   , 0<δ<1, when R(f^'/g^' (w_1))≥-2+δ/w_1   , for a some w_1∈∂D And  another, if f has angular limit at  p∈∂D . Then the inequality |(f^' (z))/(g^' (z) )|≥(-(1-2δ))/2(1+ δ)    is sharp with extremal functions  f(z)=z/(1-z)^2δ   , and   g(z)=z/(1+z), where 0<δ<1. Finally, if  f  is extended continuously to the boundary of D , then  |  (f^'/g^' )(p)|≥  |1-2δ|(|1-2δ|-2)/|p-δ |   ; 0<δ<1.