On new generalizations of Hilbert's inequality
Let \(\lambda>0\) and let \(f_i : (0,\infty)\rightarrow\mathbb R\) \((i=1,2)\) 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.
scientific article; zbMATH DE number 1506518