On Inequalities for 2F1 and Related Means

This informal working paper provides an expository survey of some inequalities and associated conjectures involving the Gaussian hypergeometric function 2F1 and closely related bivariate means. Recent as well as previously established results are presented for which the conjectures are known to hold. This basic investigation begins with the fundamental concept of a bivariate mean which is defined as a continuous function M : (0,∞) × (0,∞) → R satisfying min{x, y} ≤ M(x, y) ≤ max{x, y} for all x, y ≥ 0. Desirable properties possessed by some means include strictness: M(x, y) = x or M(x, y) = y if and only if x = y; symmetry: M(x, y) =M(y, x); and homogeneity: M(λx, λy) = λM(x, y) for λ > 0. For the purpose of investigating inequalities involving a homogeneous mean M, it suffices to consider the form M(1, r). Passing back to the more general case is then accomplished by using xM(1, y/x) =M(x, y). Simple examples of means abound. The arithmetic mean A(x, y) ≡ x+y 2 and the geometric mean G(x, y) ≡ √xy are two of the most famous (homgeneous, symmetric, and strict) means. These two are in fact special cases of the family of power means given by Aλ(x, y) ≡ ( xλ + yλ 2 )1/λ (λ 6= 0), with A0(x, y) ≡ √ xy. A standard argument can be used to show that the function λ 7→ Aλ is increasing. From this follows one of many proofs (see [9]) of the well-known arithmetic mean geometric mean inequality: