On a half-discrete Hilbert-type inequality related to hyperbolic functions

Abstract

By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper.

A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

“…x λ + y λ dx dy < π λ sin βπ f p,μ g q,ν , (1.4) where β, γ , λ > 0, β + γ = 1, μ(x) = x p(1-λβ)-1 , and ν(x) = x q(1-λγ )-1 . With regard to some other extended forms of inequalities (1.1) and (1.2), we refer to [5][6][7][8][9][10][11]. Such inequalities as (1.3) and (1.4) are commonly known as Hilbert-type inequalities.…”