New criteria for meromorphic $p$-valent starlike functions

Abstract

Let $B_{n}(\alpha)$ be the class of functions of the form $f(z)=\frac{a_{-p}}{z^{p}}+\sum_{k=0}^{\infty}$ $(a_{-p}\neq 0, p\in N=\{1,2, \cdots\})$ which are regular in the punctured disc$U^{*}=\{z:0 \lt |z| \lt 1\}$ and satisfying ${\rm Re}\{\frac{D^{n+1}f(z)}{D^{n}f(z)}-(p+1)\} \lt -\alpha$ $(n\in N_{0}=\{0,1, \cdots\}, |z| \lt 1,0\leqq\alpha \lt p)$, where $D^{n}f(z)=\frac{a_{-p}}{z^{p}}+\sum_{m=1}^{\infty}(p+m)^{n}a_{m-1}z^{m-1}$ It is proved that $B_{n+1}(\alpha)\subset B_{n}(\alpha)$. Since $B_{0}(\alpha)$ is the class of meromorphically $p$-valent starlike functions of order $\alpha$, all functions in $B_{n}(\alpha)$ are $p$-valent starlike. Further a property preserving integrals is considered.