Abstract
Using the way of weight functions and the technique of real analysis, a half-discrete Hilbert-type inequality with a general homogeneous kernel is obtained, and a best extension with two interval variables is given. The equivalent forms, the operator expressions, the reverses and some particular cases are considered. 2000 Mathematics Subject Classification: 26D15; 47A07.
Highlights
Assuming that p > 11 + = 1, pq f (≥ 0) Î Lp (R+), g(≥ Lq (R+), f p= ∞f p(x)dx p > 0, || g ||q >0, we have the following Hardy-Hilbert’s integral inequality [1]: ∞∞
On half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardy et al provided a few results in Theorem 351 of [1]
The lemma is proved. ■ Lemma 2 Let the assumptions of Lemma 1 be fulfilled and p >0(p ≠ 1), 11 p + q = 1, an ≥ 0, n ≥ n0(n Î N), f (x) is a non-negative measurable function in (b, c)
Summary
F p(x)dx p > 0, || g ||q >0, we have the following Hardy-Hilbert’s integral inequality [1]:. On half-discrete Hilbert-type inequalities with the non-homogeneous kernels, Hardy et al provided a few results in Theorem 351 of [1]. They did not prove that the constant factors are the best possible. Yang [17] gave the following half-discrete Hilbert’s inequality with the best constant factor B(l1, l2)(l, l1 >0, 0
Sign-in/Register to view highlights & summary 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.
