Abstract
By means of real analysis and weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.
Highlights
If < ∞ f (x) dx < and g (y) dy∞, we have the followingHilbert’s integral inequality:∞ ∞ f (x)g(y) dx dy < π f (x) dx g (y) dy, ( )x+y where the constant factor π is the best possible
The constant factors related to the gamma function are proved to be the best possible
5 Conclusions In this paper, by means of real analysis and weight functions a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters are obtained by Theorem
Summary
In , Hardy et al gave an extension of ( ) as follows: If k (x, y) is a non-negative homogeneous function of degree – , kp =. Φ (σ the following another kind of Hardy-type integral inequality with the non-homogeneous kernel: g(y) h(xy)f (x) dx dy y. By real analysis and the weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters as (min{xy, })α | ln xy|β (max{xy, })λ+α. > –α, there exists a constant M , such that, for any non-negative measurable functions f (x) and g(y) in ( , ∞), the following inequality:.
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