Converses of two inequalities of Ky Fan, O. Taussky, and J. Todd
Simple new proofs of the following inequalities of G. V. Milovanović and I. Ž. Milovanović are given: \[ \sum^{n+1}_{k=1}(a_ k- a_{k-1})^ 2\leq 2\left(1+\cos{\pi\over n+1}\right)\sum^ n_{k=1}a_ k^ 2, \leqno (1) \] which holds for all real numbers \(a_ 0=0\), \(a_ 1,\ldots,a_ n\), \(a_{n+1}=0\), and \[ \sum^ n_{k=1}(a_ k-a_{k-1})^ 2\leq 2\left(1+\cos{2\pi\over 2n+1}\right)\sum^ n_{k=1}a^ 2_ k, \leqno (2) \] which is valid for all real numbers \(a_ 0=0\), \(a_ 1,\ldots,a_ n\). The constants \(2\left(1+\cos{\pi\over n+1}\right)\) and \(2\left(1+\cos{2\pi\over 2n+1}\right)\) given in (1) and (2), respectively, are best possible.
discrete inequalities of Wirtinger's type
scientific article; zbMATH DE number 57296