A half-discrete Hardy-Hilbert-type inequality related to hyperbolic secant function

Bicheng Yang,Qiang Chen

doi:10.1186/s13660-015-0929-4

Abstract

By applying weight functions and technique of real analysis, a half-discrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and a best possible constant factor are given. The equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are also considered.

Highlights

If p > p + q =, f (x), g(y) ≥, f ∈ Lp(R+), g Lq(R+), f (
About the topic of half-discrete Hilbert-type inequalities with inhomogeneous kernels, Hardy et al provided a few results in Theorem of [ ], but they did not prove that the constant factors are the best possible
By applying weight coefficients and technique of real analysis, a halfdiscrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and the best possible constant factor is given, which is an extension of ( ) for λ = and a particular kernel

Summary

If p

And g q > , we have the following Hardy-Hilbert integral inequality [ ]:. where, the constant factor π sin(π /p) is the best possible. About the topic of half-discrete Hilbert-type inequalities with inhomogeneous kernels, Hardy et al provided a few results in Theorem of [ ], but they did not prove that the constant factors are the best possible. Yang [ ] gave the following half-discrete Hardy-Hilbert’s inequality with the best possible constant factor B(λ , λ ):. A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree –λ ∈ R with the best constant factor k(λ ) is obtained as follows:. By applying weight coefficients and technique of real analysis, a halfdiscrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and the best possible constant factor is given, which is an extension of ( ) for λ = and a particular kernel.

Vn p μ μ

Vn p an q