Mapping Properties of Hypergeometric Functions and Convolutions of Starlike or Convex Functions of Order α

We determine the order of convexity of hypergeometric functions z ↦ F (a, b, c, z) as well as the order of starlikeness of shifted hypergeometric functions z ↦ zF(a, b, c, z), for certain ranges of the real parameters a, b and c. As a consequence we obtain the sharp lower bound for the order of convexity of the convolution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(f*g)(z):= \sum_{n=0}^{\infty}a_nb_nz^n$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(z) =\sum_{n=0}^{\infty} a_nz^n$\end{document} is convex of order α ∈ [0,1] and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g(z) = \sum_{n=0}^{\infty}b_nz^n$\end{document} is convex of order β ∈ [0,1], and likewise we obtain the sharp lower bound for the order of starlikeness of f * g when f, g are starlike of order α, β ∈ [1/2,1], respectively. Further we obtain convexity in the direction of the imaginary axis for hypergeometric functions and for three ratios of hypergeometric functions as well as for the corresponding shifted expressions.