Inequalities and Means from a Cyclic Differential Equation

Abstract

Abstract. We prove that the solution of the cyclic initial value problem u 1 k " 1{2´u k {pu k`1`uk`2 q pk P Z{nZq, up0q " x is convergent to an equilibrium µpxqp1, . . . , 1q, and study the properties of the function x Þ Ñ µpxq and its relation to Shapiro's inequality. IntroductionIn 1954 H. S. Shapiro [6] raised the question of provingfor all n P Nzt1, 2u and all x 1 , . . . , x n ą 0. Meanwhile, after the investigations of numerous authors, (1) is known to be false in general for even n ≥ 14 and all n ≥ 25. Moreover (1) Inspired by Shapiro's inequalities, we investigated the following corresponding initial value problem, which turns out to have some interesting aspects in its asymptotic behaviour. In the sequel, let always n ≥ 3, and let all indexes of vectors in R n be understood modulo n, that is from Z n :" Z{nZ. Let D :" tx " px 1 , . . . , x n q P R n : x k ą 0 pk P Z n qu and let f : D Ñ R n be defined by f pxq " pf 1 pxq, . . . , f n pxqq "ˆ1 2´x k x k`1`xk`2˙k PZn .2010 Mathematics Subject Classification: 26D15, 34C12.