Problem related to means

Question

I am confused on proving that

If a, b, c are 3 numbers in harmonic progression

Then

${(a^n+c^n) /2} >({(a+c)/2}) ^n$

I attempted like this...

Since a, b, c are in hp so

$(a+c) /2>b$

where b is the Harmonic Mean of a and c.

But what next?

Answer 1

Assuming that $a,c\ge0$ and $n$ is a positive integer, we have that $x^n$ is a convex function for $x\ge0$.

For any convex function, $f$, $$ \frac{f(a)+f(c)}2\ge f\left(\frac{a+c}2\right) $$