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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.

MSC:
 26D15 Inequalities for sums, series and integrals of real functions
Full Text:
References:
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