I am trying to prove the inequality:
$\sum\limits_{\mathrm{cyc}} \frac{a}{2a^2+a+1}\leq\frac{3}{4}$, with $a,b,c\in\mathbb{R}^+,\ a+b+c=3$.
Now if we consider the function $f:(0,+\infty)\to\mathbb{R},\ f(x)=\frac{x}{2x^2+x+1}$ we have that this is equivalent to asking for the proof of $\frac{f(a)}{3}+\frac{f(b)}{3}+\frac{f(c)}{3}\leq f(\frac{a+b+c}{3})=f(1)=\frac{1}{4}$ which would be true by Jensen's inequality if $f$ were concave. Now, $f$ is concave on $\mathbb{R^+}$ only on $(0,1.3)$ so it seems to me that we need to prove the claim in the more general case $a,b,c \in (0,3)$ (or we should prove that $f$ has a constrained maximum at $(a,b,c)=(1,1,1)$) but I haven't been able to do so I would appreciate an hint about how to go about proving this, thanks.
(Source: E. Chen handout on inequalities)