Explicit bounds on some new nonlinear integral inequalities with delay

The purpose of the present note is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions and generalize some results of Li et al. [Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200]. The inequalities given here can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations.     

About some new integral inequalities of Wendroff type for discontinuous functions

In this article, we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables (Wendroff type) by including also inequalities with delay. We deduce new generalizations of earlier results given by R.P. Agarwal, R. Bellman, I. Bihari, B.K. Bondge, V. Lakshmikantham, S. Leela, B.G. Pachpatte for continuous and discrete functions. Furthermore, generalizations of some results for integro-sum inequalities are obtained as well.     

On integral inequalities of the Sobolev type

Summary In this paper some new integral inequalities of the Sobolev type involving many functions of many variables are established. These in turn can be used to serve as generators of other integral inequalities.     

Estimates on solutions of some new nonlinear retarded Volterra–Fredholm type integral inequalities

Some new explicit bounds on solutions to a class of new nonlinear retarded Volterra–Fredholm type integral inequalities are established, which can be used as effective tools in the study of certain integral equations. Applications examples are also indicated.     

Approximating the Riemann-Stieltjes integral in terms of generalised trapezoidal rules

Error bounds in approximating the Riemann-Stieltjes integral in terms of some new generalised trapezoidal rules are given. Applications for approximating the Riemann integral of a two-function product are provided as well.     

Some new integral inequalities of Bellman–Bihari type with delay for discontinuous functions

In this article we investigate some integral functional inequalities of Bellman–Bihari type for piecewise-continuous functions with some fixed points of discontinuity. We also prove a new analogy and generalization of results which were obtained by Bellman and Bihari to integro-sum inequalities with delay and discontinuities that do not belong to Lipschitz’s type.     

On Opial-type inequalities in two variables

Summary In this paper we establish some new Opial-type inequalities in two variables which have a wide range of applications in the study of differential and integral equations.     

On certain integral inequalities related to Opial's inequality

The aim of the present paper is to establish some new integral inequalities involving three functions and their derivatives which in the special cases yield the well known Opial inequality and some of its generalizations.     

On the identric and logarithmic means

Summary Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b ^b /a ^a ) ^1/(b–a) , fora b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b – a)/(logb – loga), fora b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) = . In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.     

On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

About new analogies of Gronwall–Bellman–Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems

In this article we present new integral Gronwall–Bellman–Bihari type inequalities for discontinuous functions (integro-sum inequalities). As applications, we investigate estimated solutions for impulsive differential systems, conditions of boundedness, stability, practical stability.     

A Hardy-type inequality in two dimensions

Necessary and sufficient conditions are given for a weighted norm inequality for the sum of two-dimensional Hardy-type integral operators with not necessarily non-negative coefficients.     

Some new nonlinear difference inequalities and their applications

In this paper, we establish some new nonlinear difference inequalities in two independent variables, which can be used as handy tools in the study of qualitative properties of solutions of certain classes of difference equations.     

A generalization of general two-point formula with applications in numerical integration

We derive a general two-point integral quadrature formula using the concept of harmonic polynomials. An improved version of Guessab and Schmeisser’s result is given with new integral inequalities involving functions whose derivatives belong to various classes of functions (_Lp$L_p$ spaces, convex, concave, bounded functions). Furthermore, several special cases of polynomials are considered, and the generalization of well-known two-point quadrature formulae, such as trapezoid, perturbed trapezoid, two-point Newton–Cotes formula, two-point Maclaurin formula, midpoint, are obtained.     

The Gauss-Green theorem and removable sets for PDEs in divergence form

Applying a very general Gauss-Green theorem established for the generalized Riemann integral, we obtain simple proofs of new results about removable sets of singularities for the Laplace and minimal surface equations. We treat simultaneously singularities with respect to differentiability and continuity.     

On some new nonlinear discrete inequalities and their applications

In this paper, some new discrete inequalities in two independent variables which provide explicit bounds on unknown functions are established. The inequalities given here can be used as tools in the qualitative theory of certain finite difference equations.     

Integral inequalities related to Hardy's inequality

Some generalizations are given of Hardy's inequality relating toL ^p -spaces. The results include many existing integral inequalities.Dedicated to Professor Janos Aczél on his 60th birthday     

Generalized Hardy,Copson, Leindler and Bennett inequalities on time scales

In this paper, we prove some new dynamic inequalities on time scales using Hölder's inequality and Keller's chain rule on time scales. These inequalities, as special cases when the time scale and when , contain some generalizations of integral and discrete inequalities due to Hardy, Copson, Leindler and Bennett.     

The first variation of the total mass of log-concave functions and related inequalities

On the class of log-concave functions on Rⁿ${\mathbb{R}}^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such a first variation, which suggest to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski’s first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski’s problem for convex bodies, and prove some partial results towards its solution.