proof some inequality by induction

Question

I got to proof the following in-equality by induction for an assignment but having a hard time.

$$ \frac{2n}{(a+b)^n} \leq \frac{1}{a^n} + \frac{1}{b^n} $$ $a,b > 0$

Thanks in adavance!

Answer 1

Jensens inequality for the convex function $f(x)=x^{-n}$ tells us that $$ \left(\frac{a+b}2\right)^{-n}=f\left(\frac{a+b}2\right)\le\frac{f(a)+f(b)}2=\frac{a^{-n}+b^{-n}}2 $$ which transforms to $$ \frac{2^{n+1}}{(a+b)^n}\le\frac1{a^n}+\frac1{b^n} $$ Hence you only need to show that $2n\le 2^{n+1}$.

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If you can't use Jensens inequality or the chain of mean inequalities, then at least you now that there is lots of space for very sloppy estimates in intermediate steps.