On a more accurate half-discrete Hilbert's inequality

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By using the way of weight coefficients and the idea of introducing parameters and by means of Hadamard's inequality, we give a more accurate half-discrete Hilbert's inequality with a best constant factor. We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses. 2000 Mathematics Subject Classification: 26D15; 47A07.

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Published 01 January 2012
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Huang and YangJournal of Inequalities and Applications2012,2012:106 http://www.journalofinequalitiesandapplications.com/content/2012/1/106
R E S E A R C HOpen Access On a more accurate halfdiscrete Hilberts inequality * Qiliang Huangand Bicheng Yang
* Correspondence: qlhuang@yeah. net Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, Peoples Republic of China
Abstract By using the way of weight coefficients and the idea of introducing parameters and by means of Hadamards inequality, we give a more accurate halfdiscrete Hilberts inequality with a best constant factor. We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses. 2000 Mathematics Subject Classification:26D15; 47A07. Keywords:weight coefficient, parameter, equivalent form, reverse, Hilberts inequality, Hadamards inequality
1 Introduction   ∞ ∞ 2 2 0,0<ad0<b<, then we have the following Ifan,bnn=1n<ann=1n wellknown Hilberts inequality (cf. [1]):   1/2 ∞ ∞∞ ∞    ambn 2 2 < πa b,(1) m n m+n n=1m=1m=1n=1
where the constant factorπis the best possible. The integral analogue of inequality   ∞ ∞ 2 2 (1) is given as follows (cf. [2]): If0<f(x)dx<and0<g(x)dx<, then 0 0   1/2 ∞ ∞∞ ∞    f(x)g(y) 2 2   dxdy< πf(x)dx g(x)dx,(2) x+y 0 00 0
where the constant factorπis the best possible. We named inequality (2) as Hilberts integral inequality. Hardy et al. [3] proved the following more accurate Hilberts inequality:   1/2 ∞ ∞∞ ∞    ambn 2 2 < πa b,(3) m n m+n1 n=1m=1m=1n=1
where the constant factorπis still the best possible. Inequalities (1)(3) are impor tant in analysis and its applications [4]. There are lots of improvements, generaliza tions, and applications of inequalities (13), for more details, refer to literatures [518].
© 2012 Huang and Yang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.