On strengthened Hardy and Pólya–Knopp's inequalities

AbstractIn this paper we prove a strengthened general inequality of the Hardy–Knopp type and also derive its dual inequality. Furthermore, we apply the obtained results to unify the strengthened classical Hardy and Pólya–Knopp's inequalities deriving them as special cases of the obtained general relations. We discuss Pólya–Knopp's inequality, compare it with Levin–Cochran–Lee's inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Pólya–Knopp type inequality