PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 34(48), pp. 109116 (1983) 

LINEAR COMBINATIONS OF REGULAR FUNCTIONS WITH NEGATIVE COEFFICIENTSG. Lakshma Reddy and K.S. PadmanabhanRamanujan Institute for Advanced Study in Mathematics University of Madras, Madras 600005, IndiaAbstract: 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. Full text of the article:
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