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 x1, x2, …, xn are positive numbers and: 12 , or 23 . Then the Shapiro inequality states that where xn+1 = x1 and xn+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… . 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 ψ is the convex hull of f(x) = e−x and g(x) = 2 / (ex + ex/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 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 where \epsilon is close to 0. Then the left-hand side is equal to, thus lower than 10 when \epsilon is small enough. The following counter-example for n = 14 is by Troesch (1985):