Linear Combinations of Regular Functions with Negative Coefficients

G. Lakshma Reddy ; K. S. Padmanabhan

Publications de l'Institut Mathématique, _N_S_34 (1983) no. 48, p. 109 .

Résumé

Let $f(z)=a_pz^p-\sum\limits_{n=1}^\infty a_{n+k}z^{n+k}$, $k\geq p\geq 1$ with $a_p>0$, $a_{n+k}\geq 0$ be regular in $E=\{z:|z|1\}$ and $F(z)=(1-\lambda)f(z)+\lambda zf'(z)$, $z\in E$ where $\lambda\geq 0$. The radius of $p$-valent starlikeness of order $\alpha, 0\leq\alpha1$, of $F$ as $f$ varies over a certain subclass of $p$-valent regular functions in $E$ is determined, and the mapping properties of $F$ in certain other situations also are discussed.