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.
Highlights
Let an, μn > 0, a = {an}∞ n=1, and p > 1
Another Hilbert-type inequality with a half-discrete kernel involving hyperbolic secant function was established by Zhong [43] in 2012
It is of interest that we present some other half-discrete inequalities involving hyperbolic functions
Summary
By constructing new kernel functions, introducing parameters, and considering coefficient refinement, reverse form, and multi-dimensional extension, a large number of new inequalities similar to (1.1) and (1.2) were established in the past several decades (see [21–31]). These newly constructed inequalities are generally called Hilbert-type inequalities. Another Hilbert-type inequality with a half-discrete kernel involving hyperbolic secant function was established by Zhong [43] in 2012 It reads n π f (x) an sech n=1 x dx < 2 f. We will establish the following half-discrete Hilbert-type inequalities with the kernels involving hyperbolic tangent and cotangent functions: an coth.
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