Generalizations of the Jensen-Steffensen and related inequalities

We present a couple of general inequalities related to the Jensen-Steffensen inequality in its discrete and integral form. The Jensen-Steffensen inequality, Slater’s inequality and a generalization of the counterpart to the Jensen-Steffensen inequality are deduced as special cases from these general inequalities.     

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.     

三阶可微函数的若干不等式

通过建立与Bullen不等式有关的积分恒等式,利用Halder不等式、Grüss不等式、Chebychev不等式,给出三阶可微函数的一些不等式.     

Generalizations of Hardy's integral inequality

The purpose of this paper is to obtain some general inequalities which contain as special cases some recently established extensions of Hardy's integral inequality.     

The general form of Hilbert’s inequality and its converses

Summary The general form of Hilbert’s inequality and its converses are established. The integral analogues of such inequalities are also given. They are significant improvements and generalizations of many known results.     

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser–Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser–Trudinger trace inequalities, sharp Adams/Moser–Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.     

离散型Carlson's不等式的一些推广及改进

In this paper, we consider Carlson type inequalities and discuss their possible improvement. First, we obtain two different types of generalizations of discrete Carlson's inequality by using the Hlder inequality and the method of real analysis, then we combine the obtained results with a summation formula of infinite series and some Mathieu type inequalities to establish some improvements of discrete Carlson's inequality and some Carlson type inequalities which are equivalent to the Mathieu type inequalities. Finally, we prove an integral inequality that enables us to deduce an improvement of the Nagy-Hardy-Carlson inequality.     

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.     

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 inequalities and convergence theorems for Choquet integrals

Fuzzy measure (or non-additive measure), which has been comprehensively investigated, is a generalization of additive probability measure. Several important kinds of non-additive integrals have been built on it. Integral inequalities play important roles in classical probability and measure theory. In this paper, we discuss some of these inequalities for one kind of non-additive integrals—Choquet integral, including Markov type inequality, Jensen type inequality, Hölder type inequality and Minkowski type inequality. As applications of these inequalities, we also present several convergence concepts and convergence theorems as complements to Choquet integral theory.     

Predictive tools in data mining and k-means clustering: Universal Inequalities

Grouping data into meaningful clusters is very important in data mining. K-means clustering is a fast method for finding clusters in data. The integral inequalities are a predictive tool in data mining and k-means clustering. Many papers have been published on speeding up k-means or nearest neighbor search using inequalities that are specific for Euclidean distance. An extended inequality related to Hölder type for universal integral is obtained in a rather general form. Previous results of Agahi et al. (Results Math, 61:179–194, 2012) are generalized by relaxing some of their requirements, thus closing the series of papers on this topic. Chebyshev’s, Hölder’s, Minkowski’s, Stolarsky’s, Jensen’s and Lyapunov’s type inequalities for the universal integral are obtained.     

半序线性空间上线性泛函的两个Gr\"{u}ss型不等式$^{}$

证明了半序线性空间上线性泛函的两个Gr\"{u}ss型不等式, 并由此给出了Karamata型积分不等式的一种推广形式, 得到了一个新的Gr\"{u}ss型积分不等式及关于傅立叶系数的两个不等式. 最后利用所得结论研究了关于矩阵及线性算子的一些Gr\"{u}ss型不等式.     

A Feng Qi type inequality for Sugeno integral

In this paper, a Feng Qi type inequality for Sugeno integral is shown. The studied inequality is based on the classical Feng Qi type inequality for Lebesgue integral. Moreover, a generalized Feng Qi type inequality for Sugeno integral is proved with several examples given to illustrate the validity of the proposed inequalities.     

Berwald type inequality for Sugeno integral

Nonadditive measure is a generalization of additive probability measure. Sugeno integral is a useful tool in several theoretical and applied statistics which has been built on non-additive measure. Integral inequalities play important roles in classical probability and measure theory. The classical Berwald integral inequality is one of the famous inequalities. This inequality turns out to have interesting applications in information theory. In this paper, Berwald type inequality for the Sugeno integral based on a concave function is studied. Several examples are given to illustrate the validity of this inequality. Finally, a conclusion is drawn and a problem for further investigations is given.     

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.     

关于曲率积分不等式的注记

本文主要研究平面卵形线的曲率积分不等式.利用积分几何中凸集的支持函数以及外平行集的性质,得到了Gage等周不等式与曲率的熵不等式的一个积分几何的简化证明;进一步地,我们得到了一个新的关于曲率积分的不等式.     

两个严格的加权Ostrowski型不等式

使用Cauchy积分不等式和Grüss不等式的变式得到两个严格的加权Ostrowski型不等式.     

On a Geometric Inequality Related to Fractional Integration

In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman’s paper. Based on it we investigate its multilinear analogue inequalities. Combining with Gressman’s work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings. Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with \(L^{p}\) bounds.     

On the Characterization of Harmonic and Subharmonic Functions via Mean-value Properties

We consider the different characterizations of harmonic and subharmonic functions in terms of their mean values in balls and on spheres. In particular, we prove a converse of an inequality of Beardon’s for subharmonic functions, and extend Rao’s integral inequalities of Harnack type between these two means in general domains.