Starlikeness of Gaussian Hypergeometric Functions Using Positivity Techniques

In this work, sufficient conditions on the sequence {an}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_n\}$$\end{document} are obtained to guarantee the starlikeness, close-to-convexity and convex in the direction of imaginary axis of the analytic function f(z)=z+∑n=2∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=z+\displaystyle \sum \nolimits _{n=2}^{\infty }a_nz^n$$\end{document} in the unit disc D.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}.$$\end{document} These results are obtained using the positivity technique of trigonometric sum as a tool. These coefficient conditions are extended to the triplet (a,b,c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,\,b,\,c)$$\end{document} to ensure that the normalized Gaussian hypergeometric function zF(a,b;c;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$zF(a,\,b;\,c;\,z)$$\end{document} is starlike. Examples are provided to compare the obtained conditions with the existing results in the literature.