Shapiro inequality


In mathematics, the Shapiro inequality is an inequality proposed by H. Shapiro in 1954.

Statement of the inequality

Suppose is a natural number and are positive numbers and:
Then the Shapiro inequality states that
where.
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.
Interior local minima of the left-hand side are always.

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for :
Then the left-hand side is equal to, thus lower than 10 when is small enough.
The following counter-example for is by Troesch :