Improvement of an inequality of Hardy containing an estimate of the size of an intermediate derivative of a function (Q1179602)

The paper deals with a sharpening of the Hardy inequality: let \(1\leq k<n\) \((n>2\) if \(k=1)\) and let \(h>0\). Then the inequality \[ \| f^{(k)}\|\leq {n-k\over n} h^{-k}\cdot {1\over 2} \| \Delta_{\pi h}(f)\| +{k\over n}\cdot h^{n-k} \| f^{(n)}\| \] holds for every function \(f\in L_ 2(-\infty,+\infty)\) which has absolutely continuous derivative \(f^{(n-1)}\) on each compact interval and \(f^{(n)}\in L_ 2(-\infty,\infty)\). Here \(\Delta_ h(f,x)=f(x+{h\over 2})-f(x-{h\over 2})\).