In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.[1]
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 .
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 [2] and [3] rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for .[4]
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 .)[5]
Interior local minima of the left-hand side are always .[6]
Counter-examples for higher n
The first counter-example was found by Lighthill in 1956, for :[7]
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)