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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.)
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