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:
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
∑ i = 1 n x i x i + 1 + x i + 2 ≥ n 2 , {\displaystyle \sum _{i=1}^{n}{\frac {x_{i}}{x_{i+1}+x_{i+2}}}\geq {\frac {n}{2}},}
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… (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 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:
x 20 = ( 1 + 5 ϵ , 6 ϵ , 1 + 4 ϵ , 5 ϵ , 1 + 3 ϵ , 4 ϵ , 1 + 2 ϵ , 3 ϵ , 1 + ϵ , 2 ϵ , 1 + 2 ϵ , ϵ , 1 + 3 ϵ , 2 ϵ , 1 + 4 ϵ , 3 ϵ , 1 + 5 ϵ , 4 ϵ , 1 + 6 ϵ , 5 ϵ ) , {\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 ϵ {\displaystyle \epsilon } is close to 0. Then the left-hand side is equal to 10 − ϵ 2 + O ( ϵ 3 ) {\displaystyle 10-\epsilon ^{2}+O(\epsilon ^{3})} , thus lower than 10 when ϵ {\displaystyle \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 ) {\displaystyle x_{14}=(0,42,2,42,4,41,5,39,4,38,2,38,0,40)} (Troesch, 1985)
References
Fink, A.M. (1998). "Shapiro's inequality". In Gradimir V. Milovanović, G. V. (ed.). Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović. Mathematics and its Applications (Dordrecht). Vol. 430. Dordrecht: Kluwer Academic Publishers. pp. 241–248. ISBN 0-7923-4845-1. Zbl 0895.26001.
External links Usenet discussion in 1999 (Dave Rusin's notes) Shapiro inequality at PlanetMath.