Close-to-convexity and starlikeness of analytic functions

Abstract

For functions $f(z)=z^p+a_{n+1}{z}^{p+1}+...$ defined on the open unit disk, the condition $\Re (f^{\prime }(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity are investigated and relevant connections with previously known results are indicated.