Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

On a reverse Mulholland-type inequality in the whole plane with general homogeneous kernel

By using the idea of introducing parameters and weight coefficients, a new reverse discrete Mulholland-type inequality in the whole plane with general homogeneous kernel is given, which is an extension of the reverse Mulholland inequality. The equivalent forms are obtained. The equivalent statements of the best possible constant factor related to several parameters and a few applied examples are presented.

On a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel

By the use of the weight coefficients, the idea of introduced parameters and the technique of real analysis, a more accurate Hilbert-type inequality in the whole plane with the general homogeneous kernel is given, which is an extension of the more accurate Hardy–Hilbert’s inequality. An equivalent form is obtained. The equivalent statements of the best possible constant factor related to several parameters, the operator expressions and a few particular cases are considered.

THE STRUCTURAL FEATURES OF HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITIES WITH ABSTRACT HOMOGENEOUS KERNEL AND ITS APPLICATIONS

In this paper, by using the theory of local fractional calculus and some techniques of real analysis, the structural characteristics of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel are studied. At the same time, the necessary and sufficient conditions for these inequalities to take the best constant factor are discussed. As an application, some best constant factor inequalities with specific kernels are obtained.

Equivalent Parameter Conditions for the Validity of Half-Discrete Hilbert-Type Multiple Integral Inequality with Generalized Homogeneous Kernel

Let Gu,v be a homogeneous nonnegative function of order λ,Kn,xm,ρ=Gnλ1,xm,ρλ2. By using the weight coefficient method, the equivalent parameter conditions and best constant factors for the validity of the following half-discrete Hilbert-type multiple integral inequality ∫ℝ+m ∑n=1∞ Kn,xm,ρanfxdx≤Ma~p,αfq,β are discussed. Finally, its applications in operator theory are discussed.

Equivalent Conditions of a Hilbert-Type Multiple Integral Inequality Holding

Let ∑i=1n1/pi=1pi>1, in this paper, by using the method of weight functions and technique of real analysis; it is proved that the equivalent parameter condition for the validity of multiple integral Hilbert-type inequality ∫R+nKx1,⋯,xn∏i=1nfixi dx1⋯dxn≤M∏i=1nfipi,αi with homogeneous kernel Kx1,⋯,xn of order λ is ∑i=1nαi/pi=λ+n−1, and the calculation formula of its optimal constant factor is obtained. The basic theory and method of constructing a Hilbert-type multiple integral inequality with the homogeneous kernel and optimal constant factor are solved.

A New Hilbert-Type Inequality with Positive Homogeneous Kernel and Its Equivalent Forms

We establish a new inequality of Hilbert-type containing positive homogeneous kernel ( min { m , n } ) λ and derive its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss its equivalent forms and give the operator expressions in some particular cases.

On a Reverse Half-Discrete Hardy-Hilbert’s Inequality with Parameters

By means of the weight functions, the idea of introduced parameters, and the Euler-Maclaurin summation formula, a reverse half-discrete Hardy-Hilbert’s inequality and the reverse equivalent forms are given. The equivalent statements of the best possible constant factor involving several parameters are considered. As applications, two results related to the case of the non-homogeneous kernel and some particular cases are obtained.