On some classes of meromorphically p-valent starlike functions with positive coefficients (Q2711259)

In this paper, the authors consider the special class \(\Sigma^*(p,\alpha)\) of functions of the form \(f(z) = az^{-p} + \sum_{k=1}^\infty a_{k+p-1} z^{k+p-1}\) which are regular and \(p\)-valent in the punctured unit disc and satisfy Re \(\{zf'(z)/f(z)\} < -\alpha\), \((0 \leq \alpha < p)\) there. Using methods adapted to this very special class from several earlier papers, the authors prove that \(f \in \Sigma^*(p, \alpha)\) if and only if \(\sum_{k=1} ^\infty (k+p-1+\alpha) a_{k+p-1} \leq \alpha (p-\alpha)\) and this result is sharp. (There are several obvious misprints in the proof of this theorem.) This theorem is then used to prove several other results about the class.