Unified Approach to Starlike and Convex Functions Involving Convolution Between Analytic Functions

Using the idea of convolution between analytic functions, we define a class UM(g,γ,b,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {UM}(g,\gamma ,b,k)$$\end{document} of analytic functions comprising of starlike and convex functions. These functions map the open unit disc on to the conic domains. We derive some sufficient conditions and then use them to define the class UM∗(g,γ,b,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {UM}^{*}(g,\gamma ,b,k)$$\end{document}. Making use of an increasing factor sequence, we discuss a subordination result. We may relate our findings with the previously known results.