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.     

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.     

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．     

Some generalizations of Hadamard’s-type inequalities through differentiability for s-convex functions and their applications

In this paper, a general form of integral inequalities of Hermite-Hadamard’s type through differentiability for s-convex function in second sense and whose all derivatives are absolutely continuous are established. The generalized integral inequalities contributes some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.     

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.     

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.     

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.     

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.     

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．     

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.     

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.     

Explicit Bounds for Some Special Integral Inequalities on Time Scales

In this paper, we investigate some nonlinear integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results, which can be used as handy tools to study the properties of certain dynamic equations on time scales, unify and extend some integral inequalities and their corresponding discrete analogues.     

On a mixed Kernel Hilbert‐type integral inequality and its operator expressions with norm

By using some real analysis techniques, we study the structural characteristics of a multi‐parameter Hilbert‐type integral inequality with the hybrid kernel and obtain some equivalent conditions for this inequality. We also consider the operator expression of the equivalent inequalities. The conclusions not only integrate some results of references but also find some new Hilbert‐type integral inequalities with simple form by choosing suitable parameter values.     

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．     

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.     

Some new delay integral inequalities and their applications

The aim of the present paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations.     

Some inequalities of Qi type for double integrals

In the paper, the authors establish some new inequalities of Qi type for double integrals on a rectangle, from which some known integral inequalities of Qi type may be derived.     

Inequalities for convex functions via Stieltjes integral

We present a method of proving inequalities for convex functions with use of Stieltjes integral. First, we show how some well-known inequalities can be obtained, and then we show how new inequalities and stronger versions of some existing results can be obtained.     

Generalized retarded integral inequalities

We prove some new retarded integral inequalities. The results generalize those in [Y.G. Sun, On retarded integral inequalities and their applications, J. Math. Anal. Appl. 301 (2) (2005) 265–275].     

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．