Shapiro inequality

In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.^[1]

Statement of the inequality[edit]

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

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 ${\displaystyle n=12}$ ^[2] and ${\displaystyle n=23}$ ^[3] rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for ${\displaystyle n=12}$.^[4]

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 ${\displaystyle g}$.)^[5]

Interior local minima of the left-hand side are always ${\displaystyle \geq {\frac {n}{2}}}$.^[6]

Counter-examples for higher n[edit]

The first counter-example was found by Lighthill in 1956, for ${\displaystyle n=20}$:^[7]

${\displaystyle x_{20}=(1+5\epsilon ,\ 6\epsilon ,\ 1+4\epsilon ,\ 5\epsilon ,\ 1+3\epsilon ,\ 4\epsilon ,\ 1+2\epsilon ,\ 3\epsilon ,\ 1+\epsilon ,\ 2\epsilon ,\ 1+2\epsilon ,\ \epsilon ,\ 1+3\epsilon ,\ 2\epsilon ,\ 1+4\epsilon ,\ 3\epsilon ,\ 1+5\epsilon ,\ 4\epsilon ,\ 1+6\epsilon ,\ 5\epsilon ),}$

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)

References[edit]

• 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.

External links[edit]

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