<s>[INST] Proving a convexity inequality 
Given $f: \mathbb{R} \to \mathbb{R}$ convex, show that:
  $$ \frac{2}{3}\left(f\left(\frac{x+y}{2}\right) + f\left(\frac{z+y}{2}\right) + f\left(\frac{x+z}{2}\right)\right) \leq f\left(\frac{x+y+z}{3}\right) + \frac{f(x) + f(y) + f(z)}{3}.$$

I have tried some ideas, such as transforming it into 
$$ f\left(\frac{x+y}{2}\right) + f\left(\frac{z+y}{2}\right) + f\left(\frac{x+z}{2}\right) - 3f\left(\frac{x+y+z}{3}\right)\\
\leq  f(x) + f(y) + f(z) - f\left(\frac{x+y}{2}\right) - f\left(\frac{z+y}{2}\right) - f\left(\frac{x+z}{2}\right) $$ (which graphically seemed intuitive) and using that such an $f$ lies above its tangents, but did not succeed… Ideas are welcome :) 
 [/INST] Let $x\geq y\geq z$.
Consider two cases.


*

*$x\geq y\geq\frac{x+y+z}{3}\geq z$.


From here we see that $2y\geq x+z$ and easy to check that
$$\left(\frac{x+y+z}{3},\frac{x+y+z}{3},\frac{x+y+z}{3},z\right)\succ\left(\frac{x+z}{2},\frac{x+z}{2},\frac{y+z}{2},\frac{y+z}{2}\right),$$ which by Karamata gives
$$3f\left(\frac{x+y+z}{3}\right)+f(z)\geq2f\left(\frac{x+z}{2}\right)+2f\left(\frac{y+z}{2}\right).$$
Thus, it's enough to prove that
$$f(x)+f(y)\geq2f\left(\frac{x+y}{2}\right),$$ which is Jensen.


*$x\geq \frac{x+y+z}{3}\geq y\geq z$.


This case is similar. Try to kill it by yourself. 
</s>