On quotients and differences of hypergeometric functions

Abstract

For Gaussian hypergeometric functions F(x) = F(a, b; c; x), a, b, c >0, we consider the quotient QF(x, y) = (F(x) + F(y))/F(z) and the difference DF(x, y) = F(x) + F(y) - F(z) for 0 < x, y <1 with z = x + y - xy. We give best possible bounds for both expressions under various hypotheses about the parameter triple (a, b; c).