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

Owa, S.; Darwish, H. E.; Aouf, M. K.

Math. Jap. 46, No. 2, 231-236 (1997).

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.

Reviewer: V.Karunakaran (Mandurai)

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

Keywords:

multivalent functions; radius of convexity