In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954. [1]
Suppose n is a natural number and x1, x2, …, xn are positive numbers and:
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 γn/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 / (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.) [5] [6]
Interior local minima of the left-hand side are always ≥n / 2. [7]
The first counter-example was found by Lighthill in 1956, for n = 20: [8]
where is close to 0. Then the left-hand side is equal to , thus lower than 10 when is small enough.
The following counter-example for n = 14 is by Troesch (1985):