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Shapiro inequality (Theorem)

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

$\displaystyle \sum_{i=1}^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2},$
where the subscripts are to be understood modulo $ n$.

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



"Shapiro inequality" is owned by Koro. [ full author list (3) | owner history (1) ]
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See Also: Nesbitt's inequality

Other names:  Shapiro's inequality
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Cross-references: Nesbitt inequalities, subscripts, odd, even, reals, integer, positive
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This is version 4 of Shapiro inequality, born on 2003-06-26, modified 2005-02-25.
Object id is 4404, canonical name is ShapiroInequality.
Accessed 6669 times total.

Classification:
AMS MSC: 26D05 (Real functions :: Inequalities :: Inequalities for trigonometric functions and polynomials)
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