Bernoulli inequality and hypergeometric functions

Hilbert Inequality and Gaussian Hypergeometric Functions

By using the integral representation of Gaussian hypergeometric function, we obtain Hilbert type inequalities with some fractional kernels and non-conjugate parameters. Such inequalities include the constant factors expressed in terms of hypergeometric functions. Further, we obtain the best possible constants for some general cases, in conjugate case.

Integral Properties of Zonal Spherical Functions, Hypergeometric Functions and Invariant

Some integral properties of zonal spherical functions, hypergeometric functions and invariant polynomials are studied for real normed division algebras.

Maclaurin’s Inequality and a Generalized Bernoulli Inequality

Maclaurin’s inequality is a natural, but nontrivial, generalization of the arithmetic-geometric mean inequality. We present a new proof that is based on an analogous generalization of Bernoulli’s inequality. Applications of Maclaurin’s inequality to iterative sequences and probability are discussed, along with a graph-theoretic version of the inequality.

A Hypergeometric Inequality

A sequence of coefficients that appeared in the evaluation of a rational integral has been shown to be unimodal. The original proof is based on a inequality for hypergeometric functions. A generalization is presented.

Hypergeometric Functions