# Shapiro inequality

Let $n$ be a positive integer and let $x_{1},\ldots,x_{n}$ be positive reals. If $n$ is even and $n\leq 12$, or $n$ is odd and $n\leq 23$, then

 $\sum_{i=1}^{n}\frac{x_{i}}{x_{i+1}+x_{i+2}}\geq\frac{n}{2},$

where the subscripts are to be understood modulo $n$.

The particular case of $n=3$ is also known as Nesbitts inequality.

Title Shapiro inequality ShapiroInequality 2013-03-22 13:43:15 2013-03-22 13:43:15 Koro (127) Koro (127) 7 Koro (127) Theorem msc 26D05 Shapiro’s inequality NesbittsInequality