Certain results of starlike and convex functions in some conditions

Abstract

The theory of geometric functions was first introduced by Bernard Riemann in 1851. In 1916, with the concept of normalized function revealed by Bieberbach, univalent function concept has found application area. Assume f(z)=z+??, n?(anzn) converges for all complex numbers z with |z|<1 and f(z)is one-to-one on the set of such z. Convex and starlike functions f(z) and g(z) are discussed with the help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)=f'(0)=1, and g(0)=0, g'(0)-1=0. A single valued function f(z) is said to be univalent (or schlict or one-to-one) in domain D?C never gets the same value twice; that is, if f(z1)-f(z2)?0 for all z1 and z2 with z1 ? z2. Let A be the class of analytic functions in the unit disk U={z:|z|<1} that are normalized with f(0)=F'(0)=1. In this paper we give the some necessary conditions for f(z) ? S* [a, a2] and 0?a2?a?1 f'(z)(2r-1)[1-f'(z)]+zf?(z / 2r[f'(z)]2. This condition means that convexity and starlikeness of the function f of order 2-r.