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) =z₂F₁ (a, b;c;z) for real parameters with 0 > a ≤ b ≤ c 0>a b c and its variant g (z) = z 2 F 2 (a, b ; c ; z 2) g (z) =z₂F₂ (a, b;c;z²). 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, + ∞) C [1, +) 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.