Nesbitt's inequality is a famous inequality with many unique solutions. It states that
Below is a list of proofs of Nesbitt's inequality. Feel free to add your own proofs.
Prove Nesbitt's Inequality
Clearing denominators and full expansion gives that it is equivalent to
However, by AM-GM
so summing the inequality symmetrically gives the desired inequality.
By Titu's lemma, it remains to prove
which is true after full expansion and the use of .
Without losing generality we assume that , which implies
Now, applying Chebyshev's inequality, we have
Applying Titu's lemma, we get , so
Let denote the left side of the inequality, then we can transform as follows:
Now, let then