Weighted Cebysev Type Inequalities for Double Integrals and Application

This article deals with two fundamental topics in mathematical analysis: the formulation of integral expressions and the derivation of integral inequalities. In particular, it introduces new one-parameter integral formulas and inequalities of the logarithmic type, where the integrands involve the logarithmic function in one way or another. Among the results are weighted Hölder-type integral inequalities and two different forms of Hardy-Hilbert-type integral inequalities. These results are illustrated by various examples and accompanied by rigorous proofs.

The aim of this paper is to establish new extension of the weighted Montgomery identity for functions of two independent variables, then obtain new Čebyšev type inequalities.

We establish two Ostrowski type inequalities for double integrals of second order partial derivable functions which are bounded. Then, we deduce some inequalities of Hermite-Hadamard type for double integrals of functions whose partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Finally, some applications in Numerical Analysis in connection with cubature formula are given.

In this research, we first provide new and refined fractional integral Mercer inequalities for harmonic convex functions by deploying the idea of line of support. Thus, these refinements allow us to develop new extensions for integral inequalities pertaining harmonic convex functions. We also provide some new fractional auxiliary equalities in Mercer sense. By employing Mercer?s harmonic convexity on them, we exhibit new fractional Mercer variants of trapezoid and midpoint type inequalities. We prove new Hermite-Hadamard (H-H) type inequalities with special functions involving fractional integral operators. For the development of these new integral inequalities, we use Power-mean, H?lder?s and improved H?lder integral inequalities. We unveiled complicated integrals into simple forms by involving hypergeometric functions. Visual illustrations demonstrate the accuracy and supremacy of the offered technique. As an application, new bounds regarding hypergeometric functions as well as special means of R (real numbers) and quadrature rule are exemplified to show the applicability and validity of the offered technique.

In this paper, we obtain some new Ostrowski type inequalities and Cebysev type inequalities for functions whose second derivatives absolute value are convex and second derivatives belongs to Lp spaces. Applications to a composite quadrature rule, to probability density functions, and to special means are also given.

This paper proposes novel single and double integral inequalities with arbitrary approximation order by employing shifted Legendre polynomials and Cholesky decomposition, and these inequalities could significantly reduce the conservativeness in stability analysis of linear systems with interval time-varying delays. The coefficients of the proposed single and double integral inequalities are determined by using the weighted least-squares method. Also former well-known integral inequities, such as Jensen inequality, Wirtinger-based inequality, auxiliary function-based integral inequalities, are all included in the proposed integral inequalities as special cases with lower-order approximation. Stability criterions with less conservatism are then developed for both constant and time-varying delay systems. Several numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.

Stochastic stability and extended dissipativity analysis for delayed neural networks with markovian jump via novel integral inequality

SummaryIntegral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Based on the extreme value conditions of a multiple variables function, a new class of Wirtinger-type double integral inequality is established in this paper. The proposed inequality generalizes and refines the classical Wirtinger-based integral inequality and has less conservatism in comparison with Jensen’s double integral inequality and other double integral inequalities in the literature. Thus, the stability criteria for delayed control systems derived by the proposed refined Wirtinger-type integral inequality are less conservative than existing results in the literature.

The main aim of this paper is to study the Ostrowski inequality for convex and (1, h, ?-m)-convex functions using the weighted Montgomery identity. Through the application of the power mean inequality, we derive results for differentiable functions by analyzing the convexity of the absolute value of their derivatives.

The key purpose of this study is to suggest a delta Riemann-Liouville (RL) fractional integral operators for deriving certain novel refinements of Polya-Szego and Cebysev type inequalities on time scales. Some new Polya-Szego, Cebysev and extended Cebysev inequalities via delta-RL fractional integral operator on a time scale that captures some continuous and discrete analogues in the relative literature. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities.

A new generalization of the weighted majorization theorem for n-convex functions is given, by using a generalization of Taylor’s formula. Bounds for the remainders in new majorization identities are given by using the Cebysev type inequalities. Mean value theorems and n-exponential convexity are discussed for functionals related to the new majorization identities.

Making use of an identity of Dragomir and Barnett, proved in [13] [published in J. Indian Math. Soc. (N.S.), 66 (1999), No. 1-4, 237-245], some new Ostrowski and Cebysev type inequalities involving two functions have been developed. Bounds obtained for the new established Ostrowski and Cebysev type inequalities are of interest and are better than the bounds available in the literature for these type of inequalitie

Improved integral inequalities for stability analysis of delayed neural networks

SummaryThis work focuses on the absolute stability problem of Lurie control system with interval time‐varying delay and sector‐bounded nonlinearity. Firstly, we present a refined Wirtinger's integral inequality and establish an improved Wirtinger‐type double integral inequality. Secondly, a modified augmented Lyapunov‐Krasovskii functional (LKF) is constructed to analyze the stability of Lurie system, where the information on the lower and upper bounds of the delay and the delay itself are fully exploited. Based on the proposed integral inequalities and some bounding techniques, the upper bound of the derivative of the LKF can be estimated more tightly. Accordingly, the proposed absolute stability criteria, formulated in terms of linear matrix inequalities, are less conservative than those in previous literature. Finally, numerical examples demonstrate the effectiveness and advantage of the proposed method.