<s>[INST] Prove inequality $2\le\frac{a+b}{c+1} +\frac{b+c}{a+1} +\frac{c+a}{b+1} \le3$ Assume that $a,b,c$ are real numbers from the interval $(\frac{1}{2},1)$. What is the proof that
$$2\le\frac{a+b}{c+1} +\frac{b+c}{a+1} +\frac{c+a}{b+1} \le3$$
holds?
 [/INST] Let $f(a,b,c)=\sum\limits_{cyc}\frac{a+b}{c+1}$.
Hence, $\frac{\partial^2f}{\partial a^2}=\frac{2(b+c)}{(a+1)^3}>0$, which says that $f$ is a convex function of $a$, of $b$ and of $c$.
Thus, 
$$\max_{\{a,b,c\}\subset\left[\frac{1}{2},1\right]}f=\max_{\{a,b,c\}\subset\{\frac{1}{2},1\}}f=f(1,1,1)=3.$$
Done!
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