Slavko Simic

Means and their inequalities

The theory of means has its roots in the work of the Pythagoreans who introduced the harmonic, geometric, and arithmetic means with reference to their theories of music and arithmetic. Later, Pappus introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among the harmonic, geometric, and arithmetic means. Nowadays, the families and types of means that are being investigated by researchers and the variety of questions that are being asked about them are beyond the scope of any single survey, with the voluminous book Handbook of Means and Their Inequalities by P. S. Bullen being the best such reference in this direction. The theory of means has grown to occupy a prominent place in mathematics with hundreds of papers on the subject appearing every year. The strong relations and interactions of the theory of means with the theories of inequalities, functional equations, and probability and statistics add greatly to its importance.

On Logarithmic Convexity for Differences of Power Means

We proved a new and precise inequality between the differences of power means. As a consequence, an improvement of Jensens inequality and a converse of Holders inequality are obtained. Some applications in probability and information theory are also given.

Landen Inequalities for Zero-Balanced Hypergeometric Functions

For zero-balanced Gaussian hypergeometric functions , , we determine maximal regions of plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each . Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.

Turan's inequality for appell polynomials

We give some necessary and sufficient conditions for the class of Appell polynomials to satisfy well-known Turans inequality. Among the other corollaries, we apply our results to some classes of orthogonal polynomials.

Bernoulli inequality and hypergeometric functions

Bernoulli type inequalities for functions of logarithmic type are given. These functions include, in particular, Gaussian hypergeometric functions in the zero-balanced case

On a Converse of Jensen's Discrete Inequality

F(a,b;a+b;x)\,.

On some intermediate mean values

We give the best possible global bounds for a form of discrete Jensens inequality. By some examples the fruitfulness of this result is shown.

On quotients and differences of hypergeometric functions

We give an extension of well-known Stolarsky means to the multivariable case in a simple and applicable way. Some basic inequalities concerning this matter are also established with applications in Analysis and Probability Theory.

On Moments of the Power Series Distributions

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class of mean values where are continuously differentiable convex functions satisfying the relation ,  . Then we asked for a characterization of such that the inequalities or hold for each positive , where are the harmonic, arithmetic, logarithmic, and identric means, respectively. For a subclass of with , this problem is thoroughly solved.