Conditions for Starlikeness of Multivalent Functions

A function f(z) meromorphic in a domain D⊂C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subset {\mathbb {C}}$$\end{document} is said to be p-valent in D if for each w the equation f(z)=w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=w$$\end{document} has at most p roots in D, where roots are counted in accordance with their multiplicity, and there is some v such that the equation f(z)=v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=v$$\end{document} has exactly p roots in D. We prove some new sufficient conditions for functions to be p-valently starlike in the unit disc.