Abstract
By introducing two pairs of conjugate exponents and using the improved Euler-Maclaurin summation formula, we estimate the weight functions and obtain a half-discrete Hilbert-type inequality with the non-monotone kernel and the best constant factor. We also consider its equivalent forms.
Highlights
If an, bn ≥, such that < ∞ n= a n < ∞ andHilbert’s inequality as follows: b n∞, we have the famous∞ ∞ ambn < π m+n a n b n, ( )
BY participated in the design of the study, and reformed the manuscript
All authors read and approved the final manuscript
Summary
In , Hardy et al [ ] established a few results on the half-discrete Hilbert-type inequalities with the non-homogeneous kernel (see Theorem ). They did not prove that the constant factors are the best possible. Yang et al [ – ] gave some half-discrete Hilbert-type inequalities and their reverses with the monotone kernels and best constant factors. Yang [ ] gave the following half-discrete Hilbert-type inequality with the nonmonotone kernel and the best constant factor :. By using the way of weight functions, we give a new half-discrete Hilberttype inequality with the non-monotone kernel as follows: an n=. The main objective of this paper is to build the best extension of ( ) with parameters and equivalent forms
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