### A beautiful generalization of the Nesbitt Inequality

I want to discuss a beautiful inequality, that is a generalization of the famous Nesbitt inequality:

(Romanian TST) For positive , prove that

Clearly, if , then we get Nesbitt’s inequality, which states that

.

This is question 14 on Mildorf’s “Olympiad Inequalities”, and its solution comprises finding a factor to multiply this expression with, almost out of thin air, and then use Cauchy Schwarz and AM-GM inequalities to prove the assertion. My solution is the following:

On interchanging and , the right hand side remains the same. However, the left hand side becomes

On adding these two inequalities, we get

Multiplying both sides by and then adding on both sides, we get

This is obviously true by Cauchy Schwarz. We will explain below how we got this expression.

Let us see what happens to in some detail. After multiplying by and adding , we get

.

EDIT: I assumed that this was obviously true. However, it is slightly non-trivial that this is true. For , the condition that should be true is that . This is true in our case above.

After adding the other terms also, we get

As pointed above, this is clearly by Cauchy-Schwarz.

Hence proved

Note: For the sticklers saying this isn’t a rigorous proof, a rigorous proof would entail us assuming that

, and then deriving a contradiction by proving that

, which is obviously false