A small note on re-defining variables to prove inequalities

by ayushkhaitan3437

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.