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 Milovanovic and Milovanovic [J. Math., Anal. Appl. 88 (1992), 378-387]. Using the same techniques, the inequality (2 sin pi/2(n + 1))4 SIGMA(k = 1)n b(k)2 less-than-or-equal-to SIGMA(k = 0)n - 1 (b(k) - 2b(k + 1) + b(k + 2)2 less-than-or-equal-to (2 + 2 cos pi/n + 1)2 SIGMA(k = 1)n b(k)2 is proved for all real numbers 0 = b0, b1, ..., b(n), b(n+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 (a(ij)) that are rotation-invariant, i.e., that obey (a(ij)) = (a(i + 1)(j + 1)). (C) 1994 Academic Press, Inc.