Shapiro inequality

Template:Short description In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.

Statement of the inequality

Suppose Template:Mvar is a natural number and x₁, x₂, …, x_n are positive numbers and:

• Template:Mvar is even and less than or equal to 12, or
• Template:Mvar is odd and less than or equal to 23.

Then the Shapiro inequality states that

$\sum _{i=1}^n\frac{x_i}{x_{i+1}+x_{i+2}}\ge \frac{n}{2},$

where x_n+1 = x₁ and x_n+2 = x₂. The special case with n = 3 is Nesbitt's inequality.

For greater values of Template:Mvar the inequality does not hold, and the strict lower bound is γ Template:Sfrac with γ ≈ 0.9891… (sequence A245330 in the OEIS).

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 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 Template:Mvar 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 Template:Mvar is the convex hull of the union of the regions above the graphs of Template:Mvar and Template:Mvar.)

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

Counter-examples for higher n

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

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

where $ϵ$ is close to 0. Then the left-hand side is equal to $10-{ϵ}^2+O({ϵ}^3)$, thus lower than 10 when $ϵ$ 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)

External links

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