Coefficient Conditions for Starlikeness of Nonnegative Order

Abstract

Sufficient conditions on a sequence $\{a_k\}$ of nonnegative numbers are obtained that ensures $f(z)={\sum }_{k=1}^{\infty }{a}_k{z}^k$ is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions.