New kinds of Hardy-Hilbert's integral inequalities (Q963811)

The paper establishes a generalized weighted Hardy-Hilbert's integral inequality, \[ \begin{multlined} \int_0^{\infty} \int_0^{\infty} \frac {f(x)g(x)}{h(x,y)\max\left\{k\left(\frac xy\right),k\left(\frac yx\right)\right\}} dxdy\\ \leq C \left(\int_0^{\infty} t^{1-\lambda}f^p(t)dt \right)^{1/p} \left( \int_0^{\infty} t^{1-\lambda} g^q(t)dt \right)^{1/q}, \end{multlined} \] where \(h\) is homogeneous of degree \(\lambda\) and \(k\) is non-decreasing. A value of the constant \(C\) is given as the sum of two integrals involving \(h\) and \(k\). The technique of the proof is basically Holder's inequality and estimation of integrals. The inequality is applied to some specifically selected functions of \(h\) and \(k\), and the corresponding inequalities involving Beta and Gamma functions are obtained.