A NEW EXTENSION ON THE HARDY-HILBERT INEQUALITY

• Zhou, Yu (Department of Mathematics and Computer Science Normal College, Jishou University) ;
• Gao, Mingzhe (Department of Mathematics and Computer Science Normal College, Jishou University)
• Published : 2012.07.31

Abstract

A new Hardy-Hilbert type integral inequality for double series with weights can be established by introducing a parameter ${\lambda}$ (with ${\lambda}>1-\frac{2}{pq}$) and a weight function of the form $x^{1-\frac{2}{r}}$ (with $r$ > 1). And the constant factors of new inequalities established are proved to be the best possible. In particular, for case $r$ = 2, a new Hilbert type inequality is obtained. As applications, an equivalent form is considered.

References

1. M. Gao, On Hilbert's inequality and its applications, J. Math. Anal. Appl. 22 (1997), no. 1, 316-323.
2. M. Gao, A supremum on Hardy-Hilbert's inequality with weight, J. Math. Study 31 (1998), no. 1, 18-23.
3. M. Gao and L. Hsu, A survey of various refinements and generalizations of Hilbert's inequalities, J. Math. Res. Exposition 25 (2005), no. 2, 227-243.
4. M. Gao and B. Yang, On the extended Hilbert's inequality, Proc. Amer. Math. Soc. 126 (1998), no. 3, 751-759. https://doi.org/10.1090/S0002-9939-98-04444-X
5. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2000.
6. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge, Cambridge Univ. Press, 1952.
7. Y. M. Jin, Table of Applied Integrals, Hefei, University of Science and Technology of China Press, 2006.
8. J. Kuang and L. Debnath, On new generalizations of Hilbert's inequality and their applications, J. Math. Anal. Appl. 245 (2000), no. 1, 248-265. https://doi.org/10.1006/jmaa.2000.6766
9. B. Yang, The Norm of Operator and Hilbert-Type Inequalities, Beijing, Science Press, 2009.