## Meromorphic multivalent functions with positive and fixed second coefficients.(English)Zbl 0895.30009

The authors define the following classes of analytic multivalent functions in the puctured disc $$U^*= \{z\in\mathbb{C}\mid 0<| z|< 1\}$$.
(1) $$\Sigma(p)$$ denotes the class of functions $f(z)= z^{-p}+ \sum_{n=1}^\infty a_{p+n-1} z^{p+n-1} \qquad (a_{p+n-1}\geq 0;\;p\in \mathbb{N}= \{1,2,3,\dots\})$ which are analytic and $$p$$-valent in $$U^*$$.
(2) For $$A,B$$ fixed with $$-1\leq A<B\leq 1$$, $$A+B>0$$ and $$\alpha\in [0,p)$$, $$Q^*[p,\alpha, A,B]$$ is the subclass of $$\Sigma(p)$$ satisfying $\Biggl| \frac{p+(zf'(z)/f(z))} {Bzf'(z)/ f(z)+ [pB+(A-B)(p-\alpha)]} \Biggr|<1 \qquad (z\in U^*).$ (3) $$Q_k^*[p,\alpha, A,B]$$ $$(k\in\mathbb{N})$$ denotes the subclass of $$Q^*[p,\alpha, A,B]$$ with the representation $f(z)= z^{-p}+ \sum_{n+1}^\infty a_{p+n-1} z^{p+n-1}$ where $$a_p= [(B-A) (p-\alpha)k] [2p+ 2\alpha B+(B-A) (p-\alpha)]^{-1}$$. It is shown (using known techniques) that $$Q_k^* [p,\alpha,A,B]$$ is closed under finite convex linear combinations. The radius of convexity for this class is also obtained.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions of one complex variable