- Open Access
A half-discrete Hilbert-type inequality with the non-monotone kernel and the best constant factor
© Xin and Yang; licensee Springer 2012
- Received: 19 December 2011
- Accepted: 9 August 2012
- Published: 30 August 2012
By introducing two pairs of conjugate exponents and using the improved Euler-Maclaurin summation formula, we estimate the weight functions and obtain a half-discrete Hilbert-type inequality with the non-monotone kernel and the best constant factor. We also consider its equivalent forms.
- Hilbert-type inequality
- conjugate exponent
- Hölder’s inequality
- best constant factor
- equivalent form
where the constant factor π is the best possible.
where the constant factor is the best possible. And Yang  gave the integral analogues of (2).
In 1934, Hardy et al.  established a few results on the half-discrete Hilbert-type inequalities with the non-homogeneous kernel (see Theorem 351). But they did not prove that the constant factors are the best possible. However, Yang  gave a result by introducing an interval variable and proved that the constant factor is the best possible. Recently, Yang et al. [5–9] gave some half-discrete Hilbert-type inequalities and their reverses with the monotone kernels and best constant factors.
Obviously, for a half-discrete Hilbert-type inequality with the monotone kernel, it is easy to build the relating inequality by estimating the series form and the integral form of weight functions. However, for a half-discrete Hilbert-type inequality with the non-monotone kernel, it is much more difficult to prove.
where the constant factor is the best possible. The main objective of this paper is to build the best extension of (4) with parameters and equivalent forms.
Hence, for , we have , it follows . The lemma is proved. □
The lemma is proved. □
Then we have (13). The lemma is proved. □
where the constant factors, , are the best possible.
Hence we have (15), which is equivalent to (14).
Hence we have (16), which is equivalent to (14). Therefore (14), (15) and (16) are equivalent.
By (11) and (21), we have and for , by Fatou lemma , we have . This is a contradiction. Hence we can conclude that the constant in (14) is the best possible. If the constant factors in (15) and (16) are not the best possible, then we can imply a contradiction that the constant factor in (14) is not the best possible by (17) and (19). The theorem is proved. □
Remark For , (14) reduces to (4). Inequality (4) is a new basic half-discrete Hilbert-type inequality with the non-monotone kernel.
This work was supported by the Emphases Natural Science Foundation of Guangdong Institutions of Higher Learning, College and University (No. 05Z026) and the Natural Science Foundation of Guangdong (7004344).
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