Hilbert's Inequality Research Articles

Generalization of Hilbert Inequality with Some Parameters

Abstract In this paper, by introducing some parameters and estimating the weight coefficients, some new inequalities are established. All results of Hilbert, Hardy and Yang are considered as special cases of the new given inequalities.

A generalization of Hilbert's inequality

In a generalization of the classical Hilbert inequality by Hardy, Littlewood and Polya, the best constant for an inequality is determined provided that the generating function for the corresponding matrix satisfies certain monotonicity condition. In this paper, we determine the best constant for a class of inequalities when the monotonicity condition is no longer satisfied.

A generalization of the Hilbert's type inequality

A generalization of the Hilbert's type inequality

A reverse Hilbert-like optimal inequality

We prove an inequality on positive real numbers, that looks like a reverse to the well-known Hilbert inequality, and we use some unusual techniques from Fourier analysis to prove that this inequality is optimal.

Inequalities with conjugate exponents in grand Lebesque spaces

In this paper we want to show the validity of the generalized doubleparametric Hilbert inequality with conjugate exponents in the framework of grand Lebesgue spaces as a particular case of a more general result involving an homogeneous kernel. We also study the boundedness of an integral operator with the aforementioned kernel. 2000 AMS Classification: Primary 46E30; Secondary 26D15

A Reverse Hilbert's Type Inequality

In this paper, by using the Euler-Maclaurin expansion, we establish an inequality of a weight coefficient. Using this inequality, we derive a reverse Hilbert's type inequality. As applications, an equivalent form is obtained.

An Operator Inequality Related to Hilbert's Type Inequality

Using the weight coefficient and the theory of operators, we define a new Hilbert-type operator and obtain its norm. At last, equivalent inequalities with the best constant factor and some particular norms are considered for applications.

LOCAL SUPREMA OF DIRICHLET POLYNOMIALS AND ZEROFREE REGIONS OF THE RIEMANN ZETA-FUNCTION

AbstractA new family of zerofree region of the Riemann Zeta-function ζ is identified by using Turán's (P. Turán, Eine neue Methode inter Analysis und deren Anwendungen (Akadémiai Kiadó, Budapest, Hungary, 1953); Analytic number theory, Proc. Symp. Pure Math., vol. XXIV (Amer. Math. Soc. Providence, RI, 1972)) localization criterion linking zeros of ζ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials is established by preliminary considering their local increment properties by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Turán's localization criterion allows to conclude.

On a sharper form of half-discrete Hilbert inequality

In this paper we introduce a general inequality by using the half-discrete Hilbert inequality. As an application we give a sharper form of the half-discrete Hilbert inequality and some carlson's type inequalities.

Two new forms of half-discrete Hilbert inequality

In this paper, we introduce two new forms of the half-discrete Hilbert inequality. The first form is a sharper form of the half-discrete Hilbert inequality and is related to Hardy inequality. In the second one, we give a differential form of this inequality.

On half-discrete Hilbert’s inequality

In this paper we provide an application of the Euler–Maclaurin summation formula with the Bernoulli function for the proof of a strengthened version of the half-discrete Hilbert inequality with the best constant factor in terms of the Euler–Mascheroni constant. Some equivalent numerical representations, operator representations, two kinds of reverses as well as an extension in terms of parameters and the Beta function are also studied.

A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

Hilbert-type inequalities for Hilbert space operators

In this paper we establish a general form of the Hilbert inequality for positive invertible operators on a Hilbert space. Special emphasis is given to such inequalities with homogeneous kernels. In some general cases the best possible constant factors are also derived. Finally, we obtain the improvement of previously deduced results, based on the application of the Hermite-Hadamard inequality.

A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

In this paper,we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

On Some Generalizations of the Hilbert-Hardy Type Integral Inequalities

In this paper, new generalization of Hilbert's integral inequality is presented by using new techniques in proof. By the weight coefficient functions in the hypothesis and the parameters, we obtained many special cases which include, in particular, the integral inequalities derived by Hilbert, Hardy and Salem. As an application of our results, we construct some examples that ensure our results.

A Simple Elementary Proof of Hilbert's Inequality

We give a very simple proof of Hilbert's inequality.

A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

In this paper, we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

Inequalities Similar to Hilbert's Inequality

In the present paper, we establish some new inequalities similar to Hilbert’s type inequalities. Our results provide some new estimates to these types of inequalities.

A More Accurate Half-Discrete Hilbert Inequality with a Nonhomogeneous Kernel

By using the way of weight functions and the idea of introducing parameters, a more accurate half-discrete Hilbert inequality with a nonhomogeneous kernel and a best constant factor is presented. We also consider its best extension with the parameters, the equivalent forms, and the operator expressions as well as some reverses.

A Hilbert integral inequality with Hurwitz zeta function

They are the famous Hilbert integral inequalities, where the constant factors π2 and π are the best possible. And the equalities in (1.1) and (1.2) hold if and only if f (x) = 0, or g(x) = 0. These results can be found in papers [1] and [2]. Owing to the importance of the Hilbert inequality and the Hilbert type inequality in Mathematical analysis and applications, some mathematicians have been studying them. Recently, various refinements, extensions and generalizations of (1.2) appear in a great deal of the articles (such as [3]–[10] etc.). However, the research articles of (1.1) are few. The aim of the present paper is to build an inequality of the form