Universal convexity and range problems of shifted hypergeometric functions

In the present paper, we study the shifted hypergeometric function $f(z)=z\text{Gauss}(a,b;c;z)$ for real parameters with $0<a\le b\le c$ and its variant $g(z)=z\text{Gauss}(a,b;c;z^2)\mathrm{.}$ Our first purpose is to solve the range problems for $f$ and $g$ posed by Ponnusamy and Vuorinen in their 2001 paper. Ruscheweyh, Salinas and Sugawa developed in their 2009 paper the theory of universal prestarlike functions on the slit domain $\text{C}\setminus [1,+\mathrm{\infty })$ and showed universal starlikeness of $f$ under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case $b=1.$ Our second purpose is to show universal convexity of $f$ under certain conditions on the parameters.