Shapiro inequality

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

Statement of the inequality
Suppose $$n$$ is a natural number and $$x_1, x_2, \dots, x_n$$ 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


 * $$\sum_{i=1}^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}$$

where $$x_{n+1}=x_1$$ and $$x_{n+2}=x_2$$.

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

The initial proofs of the inequality in the pivotal cases $$n=12$$ and $$n=23$$ rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for $$n=12$$.

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

Interior local minima of the left-hand side are always $$\ge\frac{n}{2}$$.

Counter-examples for higher n
The first counter-example was found by Lighthill in 1956, for $$n=20$$:
 * $$x_{20} = (1+5\epsilon,\ 6\epsilon,\ 1+4\epsilon,\ 5\epsilon,\ 1+3\epsilon,\ 4\epsilon,\ 1+2\epsilon,\ 3\epsilon,\ 1+\epsilon,\ 2\epsilon,\ 1+2\epsilon,\ \epsilon,\ 1+3\epsilon,\ 2\epsilon,\ 1+4\epsilon,\ 3\epsilon,\ 1+5\epsilon,\ 4\epsilon,\ 1+6\epsilon,\ 5\epsilon),$$

where $$\epsilon$$ is close to 0. Then the left-hand side is equal to $$10 - \epsilon^2 + O(\epsilon^3)$$, thus lower than 10 when $$\epsilon$$ is small enough.

The following counter-example for $$n=14$$ is by Troesch (1985):
 * $$x_{14} = (0, 42, 2, 42, 4, 41, 5, 39, 4, 38, 2, 38, 0, 40)$$ (Troesch, 1985)