Discrete inequalities of Wirtinger's type for higher differences (Q1372375)

In the paper, discrete Wirtinger's type inequalities of the form \[ \sum^{u_m}_{k= s_m} (\Delta^mx_k)^2\leq 4^m \cos^{2m} \Biggl({\pi\over 2n}\Biggr) \sum^n_{k= 1} x^2_k,\tag{\(*\)} \] \[ \sum^{u_m}_{k= s_m} (\Delta^mx_k)^2\geq 4^m \sin^{2m} \Biggl({\pi\over 2n}\Biggr) \sum^n_{k= 1} x^2_k,\tag{\(**\)} \] where \(s_m= 1-[m/2]\), \(u_m= n-[m/2]\), \(x_1,\dots, x_n\) are given real numbers such that \(x_t= x_{1- t}\), \(x_{n+ 1-t}= x_{n+ t}\) for \(s_m\leq t\leq 0\), and \(\sum^n_{k= 1} x_k= 0\) (in the case \((**)\)) are presented. For earlier results \((m=2)\) see, e.g., \textit{K. Fan}, \textit{O. Taussky} and \textit{J. Todd} [Monats. Math. 59, 73-90 (1955; Zbl 0064.29803)] and \textit{W. Chen} [Arch. Math. 62, No. 4, 315-320 (1994; Zbl 0791.26011)].