Shapiro inequality

In mathematics, the Shapiro inequality is an inequality due to H. Shapiro and Vladimir Drinfel'd.

tatement of the inequality

Suppose "n" is a natural number and $x\_1,\; x\_2,\; dots,\; x\_n$ 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

:$sum\_\{i=1\}^\{n\}\; frac\{x\_i\}\{x\_\{i+1\}+x\_\{i+2\; geq\; frac\{n\}\{2\}$

where $x\_\{n+1\}=x\_1,\; x\_\{n+2\}=x\_2$.

For greater values of "n" the inequality does not hold and the strict lower bound is $gamma\; frac\{n\}\{2\}$ with $gamma\; approx\; 0.9891...$.

This result was shown by Vladimir Drinfel'd, who won a Fields Medal in 1990. Specifically, Drinfel'd showed that the strict lower bound "γ" is given by $frac\{1\}\{2\}\; psi(0)$, where "ψ" is the function convex hull of "f"("x") = "e"^−"x" and $g(x)\; =\; frac\{2\}\{e^x+e^\{frac\{x\}\{2\}$

External links

* [http://planetmath.org/encyclopedia/ShapiroInequality.html PlanetMath]