# Shapiro inequality

In mathematics, the Shapiro inequality is an inequality proposed by H. Shapiro in 1954.

## Statement of the inequality

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

• n is even and less than or equal to 12, or
• n is odd and less than or equal to 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 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 n = 12 (Godunova and Levin, 1976) and n = 23 (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for n = 12.

The value of γ was determined in 1971 by Vladimir Drinfeld, who won a Fields Medal in 1990. Specifically, Drinfeld showed that the strict lower bound γ is given by ${\displaystyle {\frac {1}{2}}\psi (0)}$, where ψ is the function convex hull of f(x) = ex and ${\displaystyle g(x)={\frac {2}{e^{x}+e^{\frac {x}{2}}}}}$. (That is, the region above the graph of ψ is the convex hull of the union of the regions above the graphs of f and g.)

Interior local mimima of the left-hand side are always ≥ n/2 (Nowosad, 1968).

## Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for 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 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)

## References

• Fink, A.M. (1998). "Shapiro's inequality". In Gradimir V. Milovanović, G. V. Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović. Mathematics and its Applications (Dordrecht). 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 n ≤ 12, from which the result for all n ≤ 12 follows. They state n = 23 as an open problem.