# zbMATH — the first resource for mathematics

On meromorphic $$p$$-valent functions with positive coefficients. (English) Zbl 1006.30011
Summary: Let $$\Sigma(p)$$ denote the class of functions of the form $f(z)=\frac{a_{p-1}}{z^p}+\sum\limits_{n=1}^{p}a_{p+n-1}z^{p+n-1},$ $$a_{p-1}>0$$, $$a_{p+n-1}\geq 0$$, $$p\in {\mathbb N}=\{1,2,\dots\}$$, which are regular and $$p$$-valent in the punctured disc $$U^*=\{z\mid 0<|z|\}$$. Let $$\Sigma_i(p)$$ ($$i\in \{0,1\}$$) denote the subclass of $$\Sigma(p)$$ satisfying $$z_0^pf(z_0)=1$$ and $$-z_0^{p+1}f'(z_0)=p$$, respectively. In this paper we obtain coefficient estimates, a distortion theorem, closure theorems and a radius of convexity of order $$\delta$$ ($$0\leq \delta<p$$) for certain subclasses $$\Sigma_iS(p,\alpha, z_0)$$, $$i\in \{0,1\}$$ of $$\Sigma_i(p)$$, $$i\in \{0,1\}$$.
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
meromporhic function; $$p$$-valent functions
Full Text: