In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.
Statement of the inequality
Suppose is a natural number and are positive numbers and:
is even and less than or equal to , or is odd and less than or equal to .
Then the Shapiro inequality states that
where and . The special case with is Nesbitt's inequality.
For greater values of the inequality does not hold, and the strict lower bound is with .
The initial proofs of the inequality in the pivotal cases and rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for .
The value of was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound is given by , where the function is the convex hull of and . (That is, the region above the graph of is the convex hull of the union of the regions above the graphs of and .)
Interior local minima of the left-hand side are always .
Similar to Nesbitt’s inequality, there is no corresponding upper bound as the cyclic sum inequality can be made arbitrarily large.
Counter-examples for higher n
The first counter-example was found by Lighthill in 1956, for :
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 is by Troesch (1985): (Troesch, 1985)
References
External links Usenet discussion in 1999 (Dave Rusin's notes)