Shapiro inequality

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

Statement of the inequality

Suppose n is a natural number and x₁, x₂, …, 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 x_n+1 = x₁ and x_n+2 = x₂. The special case with n = 3 is Nesbitt's inequality.

For greater values of n the inequality does not hold, and the strict lower bound is γ ⁠n/2⁠ with γ ≈ 0.9891… (sequence A245330 in the OEIS).

The initial proofs of the inequality in the pivotal cases n = 12^[2] and n = 23^[3] rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for n = 12.^[4]

The value of γ was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound γ is given by ψ(0), where the function ψ is the convex hull of f(x) = e^−x and g(x) = 2 / (e^x + e^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.)^[5][6]

Interior local minima of the left-hand side are always ≥ n / 2.^[7]

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for n = 20:^[8]

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

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)