Abstract
Abstract In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.
Highlights
Let ∥f ∥p,μ denote the norm of a measurable function f : + → + with respect to a measurable weighted function μ: + → +, that is
We will construct a new kernel including both the homogeneous and non-homogeneous cases, and establish a new Hilbert-type inequality which is a unified extension of inequality (1.1)–(1.9)
The discussions will be closed with some corollaries addressing special Hilbert-type inequalities with the constant factors associated with the higher derivatives of trigonometric functions
Summary
Let ∥f ∥p,μ denote the norm of a measurable function f : + → + with respect to a measurable weighted function μ: + → +, that is, p. We have some inequalities similar to inequality (1.1), such as [1] Such inequalities as inequality (1.2) are called Hilbert-type inequalities. The following inequality is a classical extension of (1.1), which was established by Yang [2] in 2004, that is,. We will establish the following Hilbert-type inequalities with the best constant factors:. We will construct a new kernel including both the homogeneous and non-homogeneous cases, and establish a new Hilbert-type inequality which is a unified extension of inequality (1.1)–(1.9). The discussions will be closed with some corollaries addressing special Hilbert-type inequalities with the constant factors associated with the higher derivatives of trigonometric functions
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