### Proving inequalities using convex functions

I have found that I am pretty bad at finding “clever factors” for Cauchy-Schwarz, whose bounds can be known from the given conditions. However, I am slowly getting comfortable with the idea of converting the expression into a convex function, and then using the Majorization Theorem.

(Turkey) Let , and positive reals such that . Find the minimum value of

My proof: We have . Hence, using AM-GM inequality, we have .

The expression we finally get is

Consider the function . By differentiating twice, we know that for , this function is convex.

Hence, will be minimized only when , which we know from the conditions given in the question, is .

Note that

Hence, the minimum is attained when , and is equal to , which is found by substituting in the original expression.