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A new class of meromorphically multivalent functions with applications to generalized hypergeometric functions. (English) Zbl 1140.30007

In this paper the authors define a class of functions $f\left(z\right)$ that are meromorphic, normalized, analytic and $p$-valent in the punctured unit disc ${U}^{*}$ and satisfy the subordination

$\gamma \frac{{\left(f*g\right)}^{\left(m\right)}\left(z\right)}{{\left(f*h\right)}^{\left(m\right)}\left(z\right)}\prec \gamma -\frac{p\left(A-B\right)z}{1+Bz}\phantom{\rule{1.em}{0ex}}\left(z\in {U}^{*}\right),$

where $\gamma >0,$ $0\le B ${b}_{n}\ge {c}_{n}\ge 0$ $\left(n\ge p\right)$, $p\ge m$ $\left(m\in \left\{2j-1:j\in ℕ\right\}\cup \left\{0\right\}\right),$ provided that ${\left(f*h\right)}^{\left(m\right)}\left(z\right)\ne 0$ $\left(z\in {U}^{*}\right)·$ Here ${b}_{n}$ and ${c}_{n}$ are the coefficients of functions $g\left(z\right)$ and $h\left(z\right),$ respectively, ${}^{\text{'}\text{'}}{*}^{\text{'}\text{'}}$ denotes the Hadamard product (convolution) and “$\prec$” denotes subordination. The authors give coefficient estimates, distortion properties and radii of convexity and starlikeness for this class of functions together with some applications of the main result concerning generalized hypergeometric functions. At the end of the paper some remarks for further investigation are given.

##### MSC:
 30C45 Special classes of univalent and multivalent functions