Certain classes of meromorphic multivalent functions. (English) Zbl 0769.30012

Let \(\Sigma_ p\) denote the family of functions \[ f(z)=z^{-p}+ a_ 0 z^{-p+1}+ a_ 1 z^{-p+2}+\dots \] which are regular in \(\Delta\setminus\{0\}\), where \(p\) is an integer and \(\Delta=\{| z|<1\}\). For \(n>-p\) and \(f\in \Sigma_ p\) define the convolution \(D^{n+p-1} f(z)=[z^{-p}(1-z)^{-n-p}] * f(z)\) and for \(-1\leq B<A\leq 1\) let \[ C_{n,p}(A,B)=\{f:\;-z^{p-1}(D^{n+p-1} f(z))' \prec p(1+Az) (1+Bz)^{-1},\;z\in\Delta\}, \] where \(\prec\) denotes subordination. In this paper the authors obtain results such as \(C_{n+1,p}(A,B)\subset C_{n,p}(A,B)\), class preserving integral operators and sharp coefficient estimation for functions in the family \(C_{n,p}(A,B)\).


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