Let $a,b,c >0$
Prove the inequality $\displaystyle{\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{1}{a+b+c+1} \geq 1}$
I dont even know where to begin.
Only interested in hints (not solution)
Let $a,b,c >0$
Prove the inequality $\displaystyle{\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{1}{a+b+c+1} \geq 1}$
I dont even know where to begin.
Only interested in hints (not solution)
Hint :
$\frac{a}{a+1} > \frac{a}{a+b+c+1}$
$\frac{b}{b+1} > \frac{b}{a+b+c+1}$
$\frac{c}{c+1} > \frac{c}{a+b+c+1}$
We have $a+1\leq a+b+c+1$, $b+1\leq a+b+c+1$ and $c+1\leq a+b+c+1$ so $\frac 1{x+1}\geq \frac 1{1+a+b+c}$ where $x=a,b$ or $c$. Now multiply by $x$ to get $\frac x{x+1}\geq \frac x{1+a+b+c}$ and sum.