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Meromorphic multivalent functions with positive coefficients. II. (English) Zbl 0718.30009
[For part I see the author in ibid. 35, No.1, 1-11 (1990; Zbl 0705.30019).] Let $H\sp*(p;A,B)$ denote the class of functions of the form $$ f(z)=z\sp{-p}+\sum\sp{\infty}\sb{n=1}a\sb{n+p-1}z\sp{n+p-1} $$ which are regular and p-valent in $D=\{z:$ $0<\vert z\vert <1\}$ and satisfying the conditions: $$ a\sb{n+p-1}\ge 0,\quad \vert z\sp{p+1}f'(z)+p\vert \le \vert \beta z\sp{p+1}f'(z)+Ap\vert \text{ for } z\in D. $$ Sharp coefficient estimates, distortion theorems, radius of meromorphic convexity estimates and extreme points are determined for this class $H\sp*(p;A,B)$. Some convolution properties and integral transforms of functions in the class $H\sp*(p;(2\alpha -1)\beta,\beta)$, $0\le \alpha <1$, $0<\beta \le 1$, are also studied. In particular it is proved: Theorem. If $f(z)\in H\sp*(p;A,B)$, then for $0<\vert z\vert =r<1$ $$ r\sp{-p}-\frac{B-A}{1+B}r\sp p\le \vert f(z)\vert \le r\sp p+\frac{B- A}{1+B}r\sp p $$ with equality on the right holds for the function $$ f\sb p(z)=z\sp{-p}+\frac{B-A}{1+B}z\sp p.$$

30C45Special classes of univalent and multivalent functions
30C50Coefficient problems for univalent and multivalent functions