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 ) f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0 > a ≤ b ≤ c 0>a\le b\le c and its variant g ( z ) = z 2 F 2 ( a , b ; c ; z 2 ) g(z)=z_{2}F_{2}(a,b;c;z^2) . Our first purpose is to solve the range problems for f f and g 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 C ∖ [ 1 , + ∞ ) \mathbb {C}\setminus [1,+\infty ) and showed universal starlikeness of f 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 b=1 . Our second purpose is to show universal convexity of f f under certain conditions on the parameters.