Nesbitt's inequality
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In mathematics, Nesbitt's inequality is a special case of the Shapiro inequality. It states that for positive real numbers a, b and c we have:
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[edit] Proof
[edit] First proof
Starting from Nesbitt's inequality(1903)
we transform the left hand side:
Now this can be transformed into:
Division by 3 and the right factor yields:
Now on the left we have the arithmetic mean and on the right the harmonic mean, so this inequality is true.
We might also want to try to use GM for three variables.
[edit] Second proof
Suppose 
, we have that
define
The scalar product of the two sequences is maximum because of the Rearrangement inequality if they are arranged the same way, call 
and 
the vector 
shifted by one and by two, we have:
Addition yields Nesbitt's inequality.
[edit] Note
This article incorporates material from Nesbitt's inequality on PlanetMath, which is licensed under the GFDL. This article incorporates material from proof of Nesbitt's inequality on PlanetMath, which is licensed under the GFDL.
