Nesbitt's inequality

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archive posted 5 weeks ago

Prove that for all positive real numbers $a$ , $b$ and $c$

\frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} \geq \frac{3}{2} .

When does equality hold?

Posed by Alfred M. Nesbitt, 1903

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algebra inequalities

History and references

Alfred Nesbitt published the inequality in 1903, and it became a standard first exercise in olympiad inequalities. It has many short proofs. One adds $1$ to each fraction, so the left side becomes $(a + b + c)$ times the sum of the reciprocals of $b + c$ , $c + a$ and $a + b$ , and then uses the inequality between the arithmetic and harmonic means. Equality holds exactly when $a = b = c$ .

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Nesbitt's inequality

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