New Proofs and a Generalization of Inequalities of Fan, Taussky, and Todd

Discrete Fourier analysis is used to obtain simple proofs of certain inequalities about finite number sequences determined by Fan, Taussky, and Todd [Monatsh. Math. 59 (1955), 73-90] and their converses determined by Milovanović and Milovanović [J. Math., Anal. Appl.88 (1992), 378-387]. Using the same techniques, the inequality [formula] is proved for all real numbers 0=b0, b1, ..., bn, bn+1=0, which answers a question raised by Alzer [J. Math. Anal. Appl.161 (1991), 142-147]. Second, the method is used to obtain the eigenvalues and eigenvectors of matrices (aij) that are rotation-invariant, i.e., that obey (aij)=(a(i+1)(j+1)).