Nesbitt's Inequality

In mathematics, Nesbitt's inequality states that for positive real a, b and c we have:
\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.

Proof

Starting from Nesbitt's inequality
\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}
we transform the left hand side:
\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\geq\frac{3}{2}.
Now this can be transformed into:
((a+b)+(a+c)+(b+c))\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\geq 9.
Division by 3 and the right factor yields:
\frac{(a+b)+(a+c)+(b+c)}{3}\geq\frac{3}{\frac{1}{a+b}+\frac{1}{a+c}+ \frac{1}{b+c}}.
Now on the left we have the arithmetic mean and on the right the harmonic mean, so this inequality is true.

 

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