Vol. 81, No. 1, 1979

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A cyclic inequality and a related eigenvalue problem

Jerry Searcy and B. Andreas Troesch

Vol. 81 (1979), No. 1, 217–226
Abstract

A cyclic sum S(x) = Σxi(xi+1 + xi+2) is formed with the N components of a vector x, where xN+1 = x1, xN+2 = x2, and where all denominators are positive and all numerators nonnegative. It is known that the inequality S(x) N∕2 does not hold for even N 14; this result is derived in a uniform manner by considering a related algebraic eigenvalue problem. Numerical evidence is presented for the conjecture that this cyclic inequality is true for even N 12 and odd N 23.

Mathematical Subject Classification 2000
Primary: 15A39
Secondary: 26D15, 65F15
Milestones
Received: 24 October 1977
Revised: 24 October 1978
Published: 1 March 1979
Authors
Jerry Searcy
B. Andreas Troesch