A cyclic inequality and a related eigenvalue problem

Jerry Searcy and B. Andreas Troesch

Vol. 81 (1979), No. 1, 217–226

DOI: 10.2140/pjm.1979.81.217

Abstract

A cyclic sum S(x) = Σx_i∕(x_i+1 + x_i+2) is formed with the N components of a vector x, where x_N+1 = x₁, x_N+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