Abstract
By introducing several independent parameters, according to the structural symmetry of quasi-homogeneous kernels and the Hilbert-type inequality, and using the weight function method, the parameter conditions of the optimal Hilbert-type n-multiple series inequality with quasi-homogeneous kernels are discussed, and several equivalent conditions and the expression formula of the best constant factor are obtained. As applications, some special symmetric inequalities are given.
Highlights
IntroductionIn 1908, the literature [1] stated the well-known Hilbert series inequality with symmetric and −1 order homogeneous kernel m+1 n :
Suppose that a = {am} ∈ l2, b = {bn} ∈ l2
It is worth pointing out that the structural symmetry of the Hilbert-type inequality makes our treatment of each variable mi of universal significance, which is very important
Summary
In 1908, the literature [1] stated the well-known Hilbert series inequality with symmetric and −1 order homogeneous kernel m+1 n :. [6] considered the symmetric homogeneous kernel (min{m, n})λ of λ-order, and obtained the Hilbert-type series inequality of the following form: If λ1 + λ2 = λ, am ≥ 0, bm ≥ 0, (min{m, n})λambn ≤ M. N=1 m=1 m=1 n=1 and the equivalent conditions and expression formula for the best constant factor are discussed. The equivalent conditions and the expression formula of the best constant factor for quasi-homogeneous kernel have not yet been seen in the literature. By introducing matching parameters a1, a2, · · · , an, and using the Hölder’s inequality and weight coefficient method, we can obtain the following Hilbert-type multiple series inequality n. It is worth pointing out that the structural symmetry of the Hilbert-type inequality makes our treatment of each variable mi of universal significance, which is very important
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