Shapiro inequality

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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, named after H. Shapiro, states that

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

where xn + 1 = x1,xn + 2 = x2.

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, for which he 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}}}

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