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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 $x_i>0$.

  1. Is there $(x_1, \ldots ,x_n)\in {\mathbb{R}^*_+}^n$ such that $f(x_1,\ldots , x_n) < n/2$?
  2. Can we find $\inf_{x_i>0}f(x_1,\ldots , x_n)$?
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2 Answers 2

up vote 5 down vote accepted

This is discussed briefly as a generalization of Shapiro's cyclic sum inequality by J. Michael Steele in his book The Cauchy-Schwarz Master Class. He remarks that (1.) holds for $n\ge25$ and refers to this paper: P. J. Bushell, Shapiro’s “Cyclic Sums", Bull. L.M.S. (1994) 26, 564–574.

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Thanks Robin, this closes my question –  Portland Sep 5 '10 at 1:21
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If you're satisfies with this response, you should accept the answer! –  Cam McLeman Sep 7 '10 at 16:15

This is discussed in detail on the following MathWorld article: Shapiro's Cyclic Sum Constant . 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 note by Khrabrov.

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Thanks FG, the MathWorld article is very useful. –  Portland Sep 5 '10 at 1:22

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