Bounds for Hardy type differences (Q546295)

The authors revisit the classical Hardy inequality and the Pólya-Knopp inequality. The paper contains necessary basic facts about convex, log-convex as well as exponentially convex functions. Let \(A_{k}\) be an integral operator defined by \[ A_k f\left( x \right) = \frac{1}{K\left( x \right)}\int_{\Omega _2} {k \left( {x,y} \right) f\left( y \right) d \mu_{2} \left( y \right),} \] where \(k: \Omega_{1} \times \Omega_{2} \rightarrow \mathbb R\) is a general nonnegative kernel, \((\Omega_{1}, \Sigma_{1}, \mu_{1}), (\Omega_{2}, \Sigma_{2}, \mu_{2})\) are measure spaces with \(\sigma \)-finite measures and \[ K\left( x \right) = \int_{\Omega _2 } {k\left( {x,y} \right)d\mu _2 \left( y \right),} \quad x \in \Omega_1. \] Guided by the above result, improvements and reversals of new weighted Hardy-type inequalities with integral operators are stated and proved. A new Cauchy-type mean is introduced and a monotonicity property of this mean is proved. The paper is useful for researchers in the field of this type of inequalities.