For zero‐balanced Gaussian hypergeometric functions F(a, b; a + b; x), a, b > 0, we determine maximal regions of ab plane where well‐known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each x ∈ (0,1). Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.

Bernoulli type inequalities for functions of logarithmic type are given. These functions include, in particular, Gaussian hypergeometric functions in the zero-balanced case <!-- MATH $F(a,b;a+b;x).$ -->

We determine the form of the best possible global bounds for the Jensen functional on the real line. Thereby, previous results on this topic are essentially improved. Some applications in Analysis and Information Theory are also given.

We presented here a refinement of Hermite-Hadamard inequality as a linear combination of its end-points. The problem of best possible constants is closely connected with well known Simpson's rule in numerical integration. It is solved here for a wide class of convex functions, but not in general. Some supplementary results are also given.

For Gaussian hypergeometric functions F(x) = F(a, b; c; x), a, b, c > 0, we consider the quotient Q _F (x, y) = (F(x) + F(y))/F(z) and the difference D _F (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). 2010 Mathematics Subject Classification: 26D06; 33C05.

We propose several improvements of the Hermite–Hadamard inequality in the form of linear combination of its end-points and establish best possible constants. Improvements of a second order for the class $\mathsf{\Phi }\left(I\right)$ with applications in Analysis and Theory of Means are also given.