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On new generalizations of Hilbert’s inequality. (English) Zbl 0970.26009
Let $\lambda>0$ and let $f_i : (0,\infty)\rightarrow\Bbb R$ $(i=1,2)$ be functions such that $0<\int^{\infty}_0 t^{1-\lambda} f_i^2 (t) dt<\infty$. Then $$ \int^{\infty}_0 \int^{\infty}_0 \frac{f_1(x)f_2(y)}{(Ax+By)^{\lambda}} dx dy < (AB)^{-\lambda/2} B\Big(\frac{\lambda}{2}, \frac{\lambda}{2}\Big)\prod^2_{i=1} \Big(\int^{\infty}_0 t^{1-\lambda} f^2_i(t) dt\Big)^{1/2} \tag 1$$ and $$ \int^{\infty}_0 y^{\lambda-1} \Big[\int^{\infty}_0 \frac{f_1(x)}{(Ax+By)^{\lambda}} dx\Big]^2 dy < (AB)^{-\lambda} \Big[B\Big(\frac{\lambda}{2}, \frac{\lambda}{2}\Big)\Big]^2 \int^{\infty}_0 t^{1-\lambda} f^2(t) dt, \tag 2$$ where $B(\cdot,\cdot)$ is the $\beta$-function. Moreover, the inequalities (1) and (2) are equivalent and the constants appearing on their right hand sides are the best possible. This is the main result of the paper. The author also presents its discrete analogue.

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
[1] Hardy, G. H.; Littlewood, J. E.; Polya, G.: Inequalities. (1952) · Zbl 0047.05302
[2] Ke, Hu: On Hilbert’s inequality. Chinese ann. Of math. 13B, 35-39 (1992) · Zbl 0757.26020
[3] Gao, M.; Li, Tan; Debnath, L.: Some improvements on Hilbert’s integral inequality. J. math. Anal. appl. 229, 682-689 (1999) · Zbl 0920.26018
[4] Gao, M.: A note on Hilbert double series theorem. Hunan ann. Math. 12, 142-147 (1992) · Zbl 0829.26015
[5] Yang, B.: On Hilbert’s integral inequality. J. math. Anal. appl. 220, 778-785 (1998) · Zbl 0911.26011
[6] Wang, Z.; Guo, D.: An introduction to special functions. (1979)
[7] Kuang, J.: Applied inequalities. (1992)
[8] Yang, B.; Debnath, L.: On a new generalization of Hardy--Hilbert’s inequality and its applications. J. math. Anal. appl. 233, 484-497 (1999) · Zbl 0935.26009