Abstract
Using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce some special cases involving homogeneous kernel. We additionally consider some particular inequalities and operator expressions.
Highlights
If f (x), g(y) ≥ 0, ∞0 < f 2(x) dx < ∞ and 0 < g2(y) dy < ∞, we have the following well-known Hilbert integral inequality:∞ ∞ f (x)g(y) dx dy < π1 2 f 2(x) dx g2(y) dy, (1)0 0 x+y with the best possible constant factor π
5 Conclusions In the present paper, using weight functions we obtain in Theorems 1, 2 a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel and multi-parameters in the whole plane
In the form of applications, a few equivalent statements of two kinds of Hardy-type integral inequalities with the homogeneous kernel in the whole plane are deduced in Corollaries 3, 7
Summary
Theorem 2 If σ1 ∈ R, the following statements (i), (ii), and (iii) are equivalent: (i) There exists a constant M2 such that, for any f (x) ≥ 0 satisfying
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