<s>[INST] $\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$ 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.
 [/INST] 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!
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