Sufficient conditions for starlikeness of reciprocal order
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A normalized analytic function $f$ defined on the unit disk $\mathbb{D}$ is starlike of reciprocal order $\alpha$, $0\leq \alpha<1$, if $\operatorname{Re}(f(z)/(zf'(z)))>\alpha$ for all $z\in \mathbb{D}$. Such functions are starlike and therefore univalent in $\mathbb{D}$. Using the well-known Miller-Mocanu differential subordination theory, sufficient conditions involving differential inclusions are obtained for a normalized analytic function to be starlike of reciprocal order $\alpha$. Furthermore, a few conditions are derived for a function $f$ to belong to a subclass of reciprocal starlike functions, satisfying $\left\lvert f(z)/ (z f'(z)) - 1 \right\rvert < 1-\alpha$.
Abstract
A normalized analytic function $f$ defined on the unit disk $\mathbb{D}$ is starlike of reciprocal order $\alpha$, $0\leq \alpha<1$, if $\operatorname{Re}(f(z)/(zf'(z)))>\alpha$ for all $z\in \mathbb{D}$. Such functions are starlike and therefore univalent in $\mathbb{D}$. Using the well-known Miller-Mocanu differential subordination theory, sufficient conditions involving differential inclusions are obtained for a normalized analytic function to be starlike of reciprocal order $\alpha$. Furthermore, a few conditions are derived for a function $f$ to belong to a subclass of reciprocal starlike functions, satisfying $\left\lvert f(z)/ (z f'(z)) - 1 \right\rvert < 1-\alpha$.
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