Abstract
Let K(x_{1},ldots,x_{n}) satisfy \t\t\tK(x1,…,txi,…,xn)=tλλiK(t−λiλ1x1,…,xi,…,t−λiλnxn)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$K(x_{1},\\ldots,tx_{i},\\ldots,x_{n})=t^{\\lambda\\lambda _{i}}K \\bigl(t^{-\\frac{\\lambda_{i}}{\\lambda_{1}}}x_{1},\\ldots,x_{i}, \\ldots,t^{-\\frac {\\lambda _{i}}{\\lambda_{n}}}x_{n} \\bigr) $$\\end{document} for t>0. With this integral kernel, by using the method and technique of weight coefficients, the equivalent conditions and the best constant factors for the validity of Hilbert-type integral inequalities involving multiple functions are discussed. Finally, the applications of the integral inequalities are considered.
Highlights
IntroductionM is a constant, we name the following inequality a Hilbert-type integral inequality:
Let x = (x1, . . . , xn), Rn+ = {x = (x1, . . . , xn) : xi > 0 (i = 1, . . . , n)}, r > 1, f (t) ≥ 0, and α be a constant
We focus on the quasi-homogeneous integral kernels, discuss the equivalent conditions for the validity of Hilbert-type integral inequalities involving multiple functions, and obtain the expressions of the best constant factors when the inequalities are established
Summary
M is a constant, we name the following inequality a Hilbert-type integral inequality:. Xn) is said to be a quasi-homogeneous function with parameters Many good results have been obtained in the study of Hilbert-type inequalities (cf [1–24]). What is the best constant factor when the inequality holds? We focus on the quasi-homogeneous integral kernels, discuss the equivalent conditions for the validity of Hilbert-type integral inequalities involving multiple functions, and obtain the expressions of the best constant factors when the inequalities are established.
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