  In the present paper, we study the shifted hypergeometric function
$f(z)=z\Gauss(a,b;c;z)$ for real parameters with $0<a\le b\le c$ and its
variant $g(z)=z\Gauss(a,b;c;z^2).$ 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 $\C\setminus[1,+\infty)$ 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.
