# A small note on re-defining variables to prove inequalities

I just want to record my solution to the following problem, as it is different from the one given online.

For $a,b,c,d$ positive real numbers, prove that $\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\geq \frac{64}{a+b+c+d}$

This has a fairly straight forward solution using Cauchy-Schwarz inequality, which for some reason I did not think of.

The way that I solved it is that I re-defined the variables: let $a=8a', b=8b', c=16c'$ and $d=32 d'$. Then this is equivalent to proving that $\frac{1}{8}\frac{1}{a'}+\frac{1}{8}\frac{1}{b'}+\frac{1}{4}\frac{1}{c'}+\frac{1}{2}\frac{1}{d'}\geq \frac{1}{\frac{a'}{8}+\frac{b'}{8}+\frac{c'}{4}+\frac{d'}{2}}$

This is easily seen to be a consequence of Jensen’s inequality, as $\frac{1}{x}$ is a convex function for positive $x$.