$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$ I'm having trouble proving that for any $x,y,z>0$ such that $x+y+z=1$ the following inequality is true:
$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$
It seems to me that Jensen's inequality could do the trick, but I'm having trouble finding the right function and the right arguments. Any help is appreciated.

Another way.
We need to prove that
$$\sum_{cyc}\left(\frac{3}{2}-\frac{3x+1}{x+1}\right)\geq0$$ or
$$\sum_{cyc}\frac{1-3x}{x+1}\geq0$$ or
$$\sum_{cyc}\left(\frac{1-3x}{x+1}+\frac{3}{4}(3x-1)\right)\geq0$$ or
$$\sum_{cyc}(3x-1)\left(\frac{3}{4}-\frac{1}{x+1}\right)\geq0$$ or
$$\sum_{cyc}\frac{(3x-1)^2}{x+1}\geq0$$ and we are done!
