# Inequality 3

• August 6th 2010, 09:34 PM
banku12
Inequality 3
If a,b,c>0 and a+b+c=3 prove that

$\frac{1}{a^{2}-a+3}+\frac{1}{b^{2}-b+3}+\frac{1}{c^{2}-c+3}\leq 1$
• August 16th 2010, 08:40 AM
thorin
Hi,

firstly, we can notice that $f(x)=\frac{1}{x^2-x+3}$ is concave on $[0,\frac{3}{2}]$ (by 2 derivations). So, if a,b, and c are in $[0,\frac{3}{2}]$, we can apply the inegality of convexity, and the result is obvious.

Now, if at least one of the 3 numbers is $> \frac{3}{2}$. Then, the 2 others are $< \frac{3}{2}$. Let's say that $c> \frac{3}{2}$ and $a< \frac{3}{2}$ and $b< \frac{3}{2}$, for example.
Then $f(c) \leq f(\frac{3}{2})$ and $f(a) \leq f(\frac{1}{2})$ and $f(b) \leq f(\frac{1}{2})$(because $f$ has a maxima in $x=\frac{1}{2}$ and is decreasing on $[\frac{1}{2},3]$.

finally, $f(a)+f(b)+f(c) \leq 2f(\frac{1}{2})+f(\frac{3}{2})<1$

is there a mistake ?
• August 16th 2010, 10:13 AM
Also sprach Zarathustra