Differential Subordination for Starlike Functions
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A normalized analytic function, $f$ defined on the open unit disk, is starlike of order $\alpha$ if $RE(zf'(z)/f(z))>\alpha$, and is said to be reciprocal starlike of order $\alpha$ if $RE(f(z)/zf'(z))>\alpha$. Such functions are univalent and, therefore we find sufficient conditions for functions to be starlike and reciprocal starlike. We prove a general differential subordination theorem and sufficient conditions in terms of $zf'(z)/f(z)$ and $1+zf''(z)/f'(z)$ for functions to be starlike. Further, we prove sufficient conditions for the reciprocal starlikeness of functions and integral operators.
Abstract
A normalized analytic function, $f$ defined on the open unit disk, is starlike of order $\alpha$ if $RE(zf'(z)/f(z))>\alpha$, and is said to be reciprocal starlike of order $\alpha$ if $RE(f(z)/zf'(z))>\alpha$. Such functions are univalent and, therefore we find sufficient conditions for functions to be starlike and reciprocal starlike. We prove a general differential subordination theorem and sufficient conditions in terms of $zf'(z)/f(z)$ and $1+zf''(z)/f'(z)$ for functions to be starlike. Further, we prove sufficient conditions for the reciprocal starlikeness of functions and integral operators.