On Ramanujan asymptotic expansions and inequalities for hypergeometric functions

Abstract

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r 2) < (2−r 2)F(a,b;a +b;r 2) holds for all r∈(0,1).