Shapiro inequality

Suppose ${\displaystyle n}$ is a natural number and ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ are positive numbers and:

• ${\displaystyle n}$ is even and less than or equal to ${\displaystyle 12}$, or
• ${\displaystyle n}$ is odd and less than or equal to ${\displaystyle 23}$.

Then the Shapiro inequality states that

${\displaystyle \sum _{i=1}^{n}{\frac {x_{i}}{x_{i+1}+x_{i+2}}}\geq {\frac {n}{2}}}$

where ${\displaystyle x_{n+1}=x_{1},x_{n+2}=x_{2}}$.

For greater values of ${\displaystyle n}$ the inequality does not hold and the strict lower bound is ${\displaystyle \gamma {\frac {n}{2}}}$ with ${\displaystyle \gamma \approx 0.9891\dots }$.

The initial proofs of the inequality in the pivotal cases ${\displaystyle n=12}$ (Godunova and Levin, 1976) and ${\displaystyle n=23}$ (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for ${\displaystyle n=12}$.

The value of ${\displaystyle \gamma }$ was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound ${\displaystyle \gamma }$ is given by ${\displaystyle {\frac {1}{2}}\psi (0)}$, where the function ${\displaystyle \psi }$ is the convex hull of ${\displaystyle f(x)=e^{-x}}$ and ${\displaystyle g(x)={\frac {2}{e^{x}+e^{\frac {x}{2}}}}}$. (That is, the region above the graph of ${\displaystyle \psi }$ is the convex hull of the union of the regions above the graphs of '${\displaystyle f}$ and ${\displaystyle g}$.)[2]

Interior local minima of the left-hand side are always${\displaystyle \geq {\frac {n}{2}}}$ (Nowosad, 1968).

The first counter-example was found by Lighthill in 1956, for ${\displaystyle n=20}$:

${\displaystyle x_{20}=(1+5\epsilon ,\ 6\epsilon ,\ 1+4\epsilon ,\ 5\epsilon ,\ 1+3\epsilon ,\ 4\epsilon ,\ 1+2\epsilon ,\ 3\epsilon ,\ 1+\epsilon ,\ 2\epsilon ,\ 1+2\epsilon ,\ \epsilon ,\ 1+3\epsilon ,\ 2\epsilon ,\ 1+4\epsilon ,\ 3\epsilon ,\ 1+5\epsilon ,\ 4\epsilon ,\ 1+6\epsilon ,\ 5\epsilon )}$ where ${\displaystyle \epsilon }$ is close to 0.

Then the left-hand side is equal to ${\displaystyle 10-\epsilon ^{2}+O(\epsilon ^{3})}$, thus lower than 10 when ${\displaystyle \epsilon }$ is small enough.

The following counter-example for ${\displaystyle n=14}$ is by Troesch (1985):

${\displaystyle x_{14}=(0,42,2,42,4,41,5,39,4,38,2,38,0,40)}$ (Troesch, 1985)

• Fink, A.M. (1998). "Shapiro's inequality". In Gradimir V. Milovanović, G. V. (ed.). Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović. Mathematics and its Applications (Dordrecht). Vol. 430. Dordrecht: Kluwer Academic Publishers. pp. 241–248. ISBN 0-7923-4845-1. Zbl 0895.26001.
• Bushell, P.J.; McLeod, J.B. (2002). "Shapiro's cyclic inequality for even n" (PDF). J. Inequal. Appl. 7 (3): 331–348. ISSN 1029-242X. Zbl 1018.26010. They give an analytic proof of the formula for even ${\displaystyle n\leq 12}$, from which the result for all ${\displaystyle n\leq 12}$ follows. They state ${\displaystyle n=23}$ as an open problem.

• Usenet discussion in 1999 (Dave Rusin's notes)
• PlanetMath