Parameterized Yang–Hilbert-Type Integral Inequalities and Their Operator Expressions

Abstract

Applying methods of Real Analysis and Functional Analysis, we build two weight functions with parameters and provide two kinds of parameterized Yang-Hilbert-type integral inequalities with the best constant factors. Equivalent forms, the reverses, and the operator expressions are also given. In particular, the Hardytype inequalities and Hardy-type operators are studied. Additionally, a number of examples with two kinds of particular kernels are considered.

On a mixed Kernel Hilbert‐type integral inequality and its operator expressions with norm

“…previous studies [16][17][18] ); Operator characterizations of Hilbert-type inequalities (cf. Yang 19,20 ) etc. The research of Hilbert-type integral inequalities with hybrid kernels is one of the important contents too.…”

Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta function

“…where the constant factor φ(σ) is the best possible; for 0 < p < 1, 1 p + 1 q = 1, we obtain the reverse of (1.0.4) (cf. [19]).…”

“…Yang [19] studied also the equivalency of (1.0.3) and (1.0.4). In 2017, Hong [21] proved an equivalent condition between (1.0.3) and some parameters.…”

On Hardy-type integral inequalities with the gamma function

“…Additionally, an extension of (4) was given as follows: where the constant factor is the best possible ( cf. [17]). For , (8) reduces to (4).…”

“…In 2013, Yang [17] studied the equivalency between (7) and (8). In 2017, Hong [18] studied an equivalent condition between (7) with a few parameters.…”