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An improvement of the Hardy-Hilbert type integral inequalities and an application. (English) Zbl 1165.26327

Summary: It is shown that Hardy-Hilbert’s integral inequality with parameter is improved by means of a sharpening of Hölder’s inequality. A new inequality is established as following \[ \int^\infty_\alpha\int^\infty_\alpha\frac{f(x)g(y)}{(x+y+2\beta)}\,dx\,dy < \frac{\pi}{\sin(\pi/p)}\{\int^\infty_\alpha f^p(x)\,dx\}^{1/p} \cdot \{\int^\infty_\alpha g^q(x)\,dx\}^{1/q} \cdot (1-R)^m, \] where \(R=(S_p(F, h)-S_q(G, h))^2, m=\min\{1/p, 1/q\}\). As application; an extension of Hardy-Littlewood’s inequality is given.

MSC:

26D15 Inequalities for sums, series and integrals
46C99 Inner product spaces and their generalizations, Hilbert spaces