An equivalent condition to the Jensen inequality for the generalized Sugeno integral.

For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given

Jensen and Chebyshev inequalities for interval-valued functions

Integralne nejednakosti Jensena i Čebiševa uopštene su za integrale bazirane na neaditivnim merama. Prvo uopštenje dokazano je za pseudo-integral skupovno-vrednosne  funkcije, a drugo za pseudo-integral realno-vrednosne funkcije u odnosu na intervalno-vrednosnu -meru. Dokazana je i uopštena nejednakost Čebiševa za pseudo-integral realno-vrednosne  funkcije i njena dva intervalno-vrednosna oblika. Nejednakost Jensena je primenjena u principu premije, a nejednakost Čebiševa na procenu verovatnoće.Integral inequalities of Jensen and Chebyshev type are generalized for integrals based on nonadditive measures. The first generalization is proven for the pseudointegral of a set valued function and the second one for the pseudo-integral of a real-valued function with respect to the interval-valued -measure. Generalized Chebyshev inequality for the pseudo-integral of a realvalued function and its two interval-valued forms are proven. Jensen inequality is applied in the premium principle and Chebyshev inequality is applied to the probability estimation

Jensen and Chebyshev inequalities for interval-valued functions

Integralne nejednakosti Jensena i Čebiševa uopštene su za integrale bazirane na neaditivnim merama. Prvo uopštenje dokazano je za pseudo-integral skupovno-vrednosne  funkcije, a drugo za pseudo-integral realno-vrednosne funkcije u odnosu na intervalno-vrednosnu -meru. Dokazana je i uopštena nejednakost Čebiševa za pseudo-integral realno-vrednosne  funkcije i njena dva intervalno-vrednosna oblika. Nejednakost Jensena je primenjena u principu premije, a nejednakost Čebiševa na procenu verovatnoće.Integral inequalities of Jensen and Chebyshev type are generalized for integrals based on nonadditive measures. The first generalization is proven for the pseudointegral of a set valued function and the second one for the pseudo-integral of a real-valued function with respect to the interval-valued -measure. Generalized Chebyshev inequality for the pseudo-integral of a realvalued function and its two interval-valued forms are proven. Jensen inequality is applied in the premium principle and Chebyshev inequality is applied to the probability estimation

Computation of Generalized Averaged Gaussian Quadrature Rules

The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie [2] developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, the author derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. In [2], [5], [3] stable numerical procedures for computation of the corresponding averaged Gaussian rules are proposed. An analogous procedure can be applied also for a more general class of weighted averaged Gaussian rules introduced in [1]. Those results are presented in [4]. Here we we give a survey of the quoted results, which are obtained jointly with L. Reichel (Kent State Univ., OH (U.S.)

Fractional Calculus Operators and the Mittag-Leffler Function

This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others