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On the inequality $$\sum\limits_{r = 1} {x_r /(x_{r + 1} + x_{r + 2} ) \geqslant n/2} $$ and some othersand some others

On the inequality $$\sum\limits_{r = 1} {x_r /(x_{r + 1} + x_{r + 2} ) \geqslant n/2} $$ and some... On the inequality ~ffix,/(x,+l + x,+,) ~ n/2 and some others To EMIL ARTIN on his 60 th birthday by L. J. MORDSL~, (Cambridge) 1. Let Xl, X2, ..., X n, Xs+ 1 ---- X 1, Xn+ 2 ---- X 2 be non-negative real numbers. H. S. SH~I~o proposed, in 1954, in the American Mathematical Monthly the Problem. To prove that if n ~ 3 and none o] the denominators are zero, then n 1 (1) ~, z, ,-1 x,+l +x,+~ ~n. Further, equality holds only when all the denominators are equal, that is, when n is odd i I x 1 = x 2 ..... x n, and when n is even i1 xl = za -~ ... and x 2 = x 4 = .... The inequality attracted considerable attention both from its simpli- city and from the difficulty of finding a proof. This is not surprising since the inequality is not true for all n. The March 1956 number of the same journal Vol. 63, pages 191, 192 contains a counter example, due to Professor LIGHTHrLL of Manchester, when n = 20 given by taking x r ---- a r + b r s where the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

On the inequality $$\sum\limits_{r = 1} {x_r /(x_{r + 1} + x_{r + 2} ) \geqslant n/2} $$ and some othersand some others

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941955
Publisher site
See Article on Publisher Site

Abstract

On the inequality ~ffix,/(x,+l + x,+,) ~ n/2 and some others To EMIL ARTIN on his 60 th birthday by L. J. MORDSL~, (Cambridge) 1. Let Xl, X2, ..., X n, Xs+ 1 ---- X 1, Xn+ 2 ---- X 2 be non-negative real numbers. H. S. SH~I~o proposed, in 1954, in the American Mathematical Monthly the Problem. To prove that if n ~ 3 and none o] the denominators are zero, then n 1 (1) ~, z, ,-1 x,+l +x,+~ ~n. Further, equality holds only when all the denominators are equal, that is, when n is odd i I x 1 = x 2 ..... x n, and when n is even i1 xl = za -~ ... and x 2 = x 4 = .... The inequality attracted considerable attention both from its simpli- city and from the difficulty of finding a proof. This is not surprising since the inequality is not true for all n. The March 1956 number of the same journal Vol. 63, pages 191, 192 contains a counter example, due to Professor LIGHTHrLL of Manchester, when n = 20 given by taking x r ---- a r + b r s where the

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Aug 29, 2008

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