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
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
