Universal convexity and range problems of shifted hypergeometric functions.

In the present paper, we study the shifted hypergeometric function f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0<a\le b\le c and its variant g(z)=z_{2}F_{2}(a,b;c;z^2). Our first purpose is to solve the range problems for f and g posed by Ponnusamy and Vuorinen [Rocky Mountain J. Math. 31 (2001), pp. 327–353]. Ruscheweyh, Salinas and Sugawa [Israel J. Math. 171 (2009), pp. 285–304] developed the theory of universal prestarlike functions on the slit domain \mathbb {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. [ABSTRACT FROM AUTHOR]