Inf of a mutivariate function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T04:10:42Z http://mathoverflow.net/feeds/question/37746 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37746/inf-of-a-mutivariate-function Inf of a mutivariate function Portland 2010-09-04T18:18:32Z 2010-09-07T16:09:47Z <p>Let $f(x_1,\ldots , x_n) = \frac{x_1}{x_2+x_3} + \frac{x_2}{x_3+x_4} + \cdots + \frac{x_n}{x_1+x_2}$, defined for $x_i>0$.</p> <ol> <li>Is there $(x_1, \ldots ,x_n)\in {\mathbb{R}^*_+}^n$ such that $f(x_1,\ldots , x_n) &lt; n/2$?</li> <li>Can we find $\inf_{x_i>0}f(x_1,\ldots , x_n)$?</li> </ol> http://mathoverflow.net/questions/37746/inf-of-a-mutivariate-function/37748#37748 Answer by Robin Chapman for Inf of a mutivariate function Robin Chapman 2010-09-04T18:37:23Z 2010-09-04T18:37:23Z <p>This is discussed briefly as a generalization of Shapiro's cyclic sum inequality by J. Michael Steele in his book <a href="http://books.google.co.uk/books?id=7Fm3r9jcbqYC&amp;lpg=PP1&amp;dq=Steele%2520Cauchy&amp;pg=PA104#v=onepage&amp;q=Shapiro%27s&amp;f=false" rel="nofollow">The Cauchy-Schwarz Master Class</a>. He remarks that (1.) holds for $n\ge25$ and refers to this paper: P. J. Bushell, Shapiro’s “Cyclic Sums", <em>Bull. L.M.S.</em> (1994) <strong>26</strong>, 564–574.</p> http://mathoverflow.net/questions/37746/inf-of-a-mutivariate-function/37763#37763 Answer by FG for Inf of a mutivariate function FG 2010-09-04T23:22:20Z 2010-09-04T23:22:20Z <p>This is discussed in detail on the following MathWorld article: <a href="http://mathworld.wolfram.com/ShapirosCyclicSumConstant.html" rel="nofollow">Shapiro's Cyclic Sum Constant </a>. Detailed proofs of the main result (inequality holds only for even $n \le 12$ and odd $n \le 23$) can be found in the following <a href="http://olympiads.mccme.ru/lktg/2010/5/5-1en.pdf" rel="nofollow">note</a> by Khrabrov.</p>