Shapiro inequality

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

Suppose n is a natural number and ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ are positive numbers and:

Then the Shapiro inequality states that

where ${\displaystyle x_{n+1}=x_{1},x_{n+2}=x_{2}}$.

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

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