On a p-adic Hilbert-type integral operator and its applications

In this note, we deal with a $p$-adic Hilbert-type integral operator induced by a symmetric homogeneous kernel of degree $-1$ and obtain the expression of the norm of this operator. As applications, we establish some new $p$-adic Hilbert-type inequalities with best constant factors.     

On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.     

关于子区间逆向的Hilbert型积分不等式

本文研究了建立逆向Hilbert型不等式问题.引入两对共轭指数,应用权函数,实分析及参量化的方法,在积分子区间上建立逆向的具有最佳常数因子及一般齐次核的Hilbert型积分不等式,等价式及一些特殊核的不等式,推广了一些文献的结果.     

关于一个基本的Hilbert型积分不等式及其推广

利用权函数的方法,建立一个基本的、-1齐次且具有最佳常数因子的Hilbert型积分不等式,还考虑了其等价式及引入参数的最佳推广情形.     

A Unified Treatment of Half-Discrete Hilbert-Type Inequalities with a Homogeneous Kernel

In this paper we provide a unified treatment of half-discrete Hilbert-type inequalities with a general homogeneous kernel. The main results are proved for the case of non-conjugate exponents. A special emphasis is given to determining conditions under which these inequalities include the best possible constants. As an application, we consider some operator expressions closely connected to established inequalities. Finally, we also provide improvements of derived half-discrete inequalities by virtue of the Hermite-Hadamard inequality.     

一个含多参数且-4齐次核的Hilbert型积分不等式

以Hilbert不等式为代表的双线型不等式是分析学的重要不等式.应用权函数方法,引入多个参数,建立了一个新的具有最佳常数因子的-4齐次核的双线型不等式.作为应用,导出其等价形式及一些特殊结果.     

On a Hilbert-type linear series operator and its applications

This paper deals with a Hilbert-type linear series operator and its norm. Several generalizations of the Hilbert-type inequalities whose kernels are symmetric and homogeneous of the first order are presented. Included are applications of some new extended Hilbert-type inequalities with the best possible constant factors and the equivalent forms are established. Also, the reverse forms and some particular forms are obtained as special cases of the results of this paper.     

On a Hilbert-Type Integral Inequality Related to the Extended Hurwitz Zeta Function in the Whole Plane

By using techniques of real analysis and weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related the extended Hurwitz zeta function is proved to be the best possible. As applications, a few equivalent statements of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and some corollaries.

     

一个负奇数齐次核的Hilbert型积分不等式

通过估算权函数,建立一个核含多参数且为-(2n+1)(n∈N)齐次的新的Hilbert型积分不等式及其等价式,并用复分析和代数的方法论证及表述最佳常数因子.作为应用,还给出其逆向形式及一些特殊结果.     

Discrete Hilbert-type inequalities with general homogeneous kernels

The main objective of this paper is an unified treatment of discrete Hilbert-type inequalities with homogeneous kernels. At the beginning, we prove and discuss two equivalent general inequalities of such type. Further, we analyze the conditions which yield the best possible constant factors in obtained inequalities. The obtained results are then applied to various settings considering homogeneous functions of a negative real degree. In such a way, we obtain the generalizations of numerous results, previously known from the literature.     

关于一个非齐次核的Hilbert型积分算子

应用权函数的方法及实分析与泛函分析的思想技巧,定义了一个非齐次核的Hilbert型积分算子,并求出其联系范数的两个等价不等式．作为应用,还考虑了其逆形式及一些特殊核的例子．     

Integral operators with homogeneous kernels in grand Lebesgue spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ? ⁿ as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.     

Hilbert不等式的各种精化与拓广综述

在本文中,我们较全面的概述了1990年至2002年间有关Hilbert级数型与积分型不等式的种种精化与拓广工作.涉及发表于海内外的文献资料40余篇,其中我国学者的工作成果占有重要份量.文末还列举了一些有待进一步研究的公开问题.     

A Hilbert-type integral inequality whose kernel is a homogeneous form of degree −3

In this paper, by estimating the weight function, we give a new Hilbert-type integral inequality whose kernel is a homogeneous form of degree −3 with the best constant factor and the reverse form is considered.     

一个引入中间变量的一般非齐次核全平面Hilbert型积分不等式

应用实分析及权函数的方法,引入一些参数及中间变量,建立一个一般非齐次核全平面Hilbert型积分不等式的若干等价陈述.常数因子被证明是最佳的.作为应用,一个一般齐次核全平面Hilbert型积分不等式的若干等价陈述被导出.我们还考虑了一些特殊情况、算子表示及若干例子.     

On a strengthened multidimensional Hilbert-type inequality

The main objective of this paper is a study of the general refinement and converse of the multidimensional Hilbert-type inequality in the so-called quotient form. Such extensions are deduced with the help of the sophisticated use of the well-known Hölder’s inequality. The obtained results are then applied to homogeneous kernels with the negative degree of homogeneity. Also, we establish the conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weighted functions. In such a way we obtain both refinements and converses of some actual results, known from the literature.     

A half-discrete Hilbert-type inequality in the whole plane related to the Riemann zeta function

By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.     

一个核为零齐次的Hilbert型不等式

通过引入权系数,应用实分析的方法,建立一个具有最佳常数因子的零齐次核的Hilbert型不等式,同时还考虑了其等价形式及逆向形式.     

具有一般齐次核多维的半离散Hardy-Hilbert型不等式

利用权函数、转换公式和实分析技巧，给出一个具有一般齐次核和最佳常数因子的多维半离散Hardy-Hilbert型不等式，它是一个已知结果的推广.此外，还讨论了等价形式、算子表示以及几种特殊应用例子.     

On a multidimensional version of the Hilbert type inequality

The main objective of this paper is a study of some new multidimensional Hilbert type inequalities with a general homogeneous kernel. We derive a pair of equivalent inequalities, and also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered.