I'm solving a series of exercises from non-linear programming problems. In my convexity section of study, I've found this problem that I have no idea on how to solve it, can you please help me giving advices or hints?
The problem:
Show that
$$\left(\frac{x_1}{2} + \frac{x_2}{3} + \frac{x_3}{12} + \frac{x_4}{12}\right)^4 \leq \frac{x_1}{2}^4 + \frac{x_2}{3}^4 + \frac{x_3}{12}^4 + \frac{x_4}{12}^4$$
By Jensen, for convex function $f$:
$$\omega_1f(x_1)+\dots+\omega_nf(x_n)\ge(\omega_1+\dots+\omega_n)f\left(\frac{\omega_1x_1+\dots+\omega_nx_n}{\omega_1+\dots+\omega_n}\right)\tag{1}$$
In this particular case:
$$f(x)=x^4,\ \omega_1=\frac12,\ \omega_2=\frac13,\ \omega_3=\omega_4=\frac1{12},\ \omega_1+\dots+\omega_4=1\tag{2}$$
By replacing (2) into (1) you get:
$$ \frac{x_1^4}{2} + \frac{x_2^4}{3} + \frac{x_3^4}{12} + \frac{x_4^4}{12}\ge\left(\frac{x_1}{2} + \frac{x_2}{3} + \frac{x_3}{12} + \frac{x_4}{12}\right)^4 $$