Inequalities similar to certain extensions of Hilbert's inequality (Q1974223)

Let \(p\in (1,\infty)\) and \(q=p/(p-1)\). Suppose that \(f\) and \(g\) are real and absolutely continuous functions on the interval \([0,x)\) and \([0,y)\), respectively, such that \(f(0)=g(0)=0\). Then \[ \begin{aligned} \int^x_0 \int^y_0 &\frac{|f(x)g(t)|}{qs^{p-1}+pt^{q-1}} ds dt\\ &\leq K(p,q,x,y)\Big(\int^x_0 (x-s)|f'(s)|^p ds\Big)^{1/p} \Big(\int^y_0 (y-t)|g'(t)|^q dt\Big)^{1/q}, \end{aligned}\leqno(*) \] where \(K(p,q,x,y)=(pq)^{-1} x^{(p-1)/p} y^{(q-1)/q}\), \(x,y\in (0,\infty)\). This is the main result of the paper. A discrete analogue of \((*)\) is also established. Furthermore, two independent variable versions of mentioned results are given.