Stieltjes integral inequalities of Gronwall type and applications

Summary We obtain estimates for solutions of integral inequalities of Gronwall type involving Stieltjes integrals and their inverse inequalities. From these we obtain some new results for integral inequalities for Riemann integrals and functional integral inequalities. Extensions are also given to Bihari type integral inequalities.Research supported by NSERC Canada.     

On new fractional integral inequalities using Marichev–Saigo–Maeda operator

Generalizations of fractional integral inequalities were introduced by many authors. The aim of our investigation is to establish some new fractional integral inequalities using Marichev–Saigo–Maeda (MSM) fractional integral operator for convex function. Further, we obtain some more fractional integral inequalities of Grüss type using MSM operator.     

Nonlinear integral inequalities involving maxima of unknown scalar functions

Some new nonlinear integral inequalities that involve the maximum of the unknown scalar function of one variable are solved. The inequalities considered are generalizations of a classical nonlinear integral inequality of Bihari. The importance of these integral inequalities is defined by their wide applications in qualitative investigations of differential equations with “maxima”, which is illustrated by some direct applications.     

Some complements to the Jensen and Chebyshev inequalities and a problem of W. Walter

Motivated by an integral inequality conjectured by W. Walter, we prove some general integral inequalities on finite intervals of the real line. In addition to supplying new proofs of Walter's conjecture, the general inequalities furnish a reverse Jensen inequality under appropriate conditions and provide generalizations of Chebyshev's integral inequality.     

Integral Equations in Measure Spaces

In this paper, we consider a class of integral equations in measure spaces, and the corresponding integral inequalities. Special cases are Volterra type integral equations and Gronwall type integral inequalities. We give different necessary and su.cient, and only su.cient conditions which together with the Lipschitz condition imply the existence and the uniqueness of solutions of the considered integral equations. We study the successive approximations for the considered integral equations. We derive estimates for the solutions of the studied integral equations and integral inequalities. Submitted: June 20, 2000?Revised: July 10, 2001     

On some fundamental integral inequalities in two independent variables

The present paper obtains two independent variable generalizations of the integral inequalities of Gollwitzer, Langenhop, and Pachpatte. The bounds provided by these inequalities are adequate in many applications in the theory of partial differential and integral equations.     

On Beckenbach–Radó type integral inequalities

Beckenbach and Radó characterized logarithmically subharmonic functions in the plane in terms of integral inequalities involving spherical averages. In this work we generalize this result to higher dimensions and thus answer to a question raised by Beckenbach and Radó. We also consider generalizations of integral inequalities suggested by Beckenbach and Radó and discuss connections to reverse Hölder inequalities and Muckenhoupt weights.     

用广义拟Hadamard卷积定义的广义积分算子的封闭性质

首先利用广义拟-Hadamard卷积引入一个广义积分算子;其次,应用两个经典不等式和系数不等式研究广义积分算子及其特殊殊算子在某些非标准化广义解析函数类上的封闭性质.     

Pachpatte几个非线性二元积分不等式的推广

本文建立了几个ｎ元非线性积分不等式解的估计．在ｎ＝２的特殊情况改进并推广了 Ｐａｃｈｐａｔｔｅ Ｂ． Ｇ．［５,９,１０］的五个定理．举例说明了结果的应用．     

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.     

加权(C)eby(s)ev-Ostrowski型不等式

关于著名的Cebysev不等式，已有众多的研究成果．通过建立积分不等式，来建立全新的加权Cebysev型积分不等式．给予了独立的证明，并给出了此类不等式的新评价．     

常曲率平面上的逆Bonnesen型不等式

利用积分几何中估计包含测度的思想给出常曲率平面上一些新的逆Bonnesen型不等式.这些不等式在欧氏平面上为著名的Bottema不等式的改进形式与新的逆Bonnesen型不等式.     

Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated.     

Inequalities for formal power series and entire functions

We present several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized Borel transforms. They are obtained through certain limit procedures which involve the multiparameter binomial inequalities, integral inequalities for continuous functions, and weighted norm inequalities for analytic functions. Some applications to the confluent hypergeometric functions, Bessel functions, Laguerre polynomials, and trigonometric functions are discussed. Also some generalizations are given.     

On some new integral inequalities of Gronwall-Bellman-Pachpatte type

In this paper, we establish new nonlinear integral inequalities of Gronwall-Bellman-Pachpatte type. These inequalities generalize some former famous inequalities and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some nonlinear ordinary differential and integral equations. The purpose of this paper is to extend certain results which proved by Pachpatte in [Inequalities for Differential and Integral Equations, Academic Press, New York and London, 1998]. Some applications are also given to illustrate the usefulness of our results.     

赋范线性空间中的Hilbert型积分不等式

基于利用一个积分恒等式的新技巧,建立了赋范线性空间中新的Hilbert型积分不等式.这些新的结果包含了n维欧氏空间中n重积分的Hilbert型积分不等式作为其特殊情形.     

一类积分不等式的推广

对一些基本的积分不等式进行了推广，给出了含有n个无关变元的更广泛的非线性积分不等式．利用所得的不等式讨论了某些非线性积分方程解的有界性．     

A Cauchy-Khinchin integral inequality

This paper discusses some Cauchy-Khinchin integral inequalities. Khinchin [2] obtained an inequality relating the row and column sums of 0-1 matrices in the course of his work on number theory. As pointed out by van Dam [6], Khinchin’s inequality can be viewed as a generalization of the classical Cauchy inequality. Van Dam went on to derive analogs of Khinchin’s inequality for arbitrary matrices. We carry this work forward, first by proving even more than general matrix results, and then by formulating them in a way that allows us to apply limiting arguments to create new integral inequalities for functions of two variables. These integral inequalities can be interpreted as giving information about conditional expectations.     

若干有关欧阳不等式的非线性积分不等式和离散不等式

获得几个非线性的积分不等式及离散不等式．它们和近期Ｐａｃｈｐａｔｅ［１］中欧阳亮不等式［２］的推广有关．作为特殊情形还导出了一些具有幂非线性的新不等式．为说明结果的有用性，讨论了某个非线性差分方程解的有界性．     

关于Mate-Nevai型非线性积分不等式

主要目的是将Mate-Nevai型积分不等式推广到非线性情形,得到了两个具有广泛意义的Mate-Nevai型非线性积分不等式,利用所得结论讨论某些积分方程解的有界性.