Some new refined Hardy type inequalities with general kernels and measures (Q623379)

Let \(p>1\) and \(f\in L^p(0,\infty)\) be a non-negative function. Hardy's integral inequality asserts that \(\int_0^\infty (\frac{1}{x}\int_0^x f(t)\,dt)^p \,dx\leq(\frac{p}{p-1})^p\int_0^\infty f^p(x)\,dx\). The authors use the notation of superquadratic and subquadratic function with an integral operator defined on the product of two measure spaces, to obtain some refined Hardy type inequalities. The result of the paper recovers some previously known generalizations and refinements of Hardy's integral inequality, furthermore, the authors give some remarkable examples and corollaries of their main result.