LINEAR COMBINATIONS OF REGULAR FUNCTIONS WITH NEGATIVE COEFFICIENTS

G. Lakshma Reddy and K.S. Padmanabhan

Abstract: 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 $\alpha, 0\leq\alpha<1$, 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.

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