three variable inequality $\sum \frac{x}{1+2x+xy}$

Question

The problem asks to prove the following inequality for positive $a$, $b$ and $c$.

$$\dfrac{a}{ab+2a+1}+\dfrac{b}{bc+2b+1}+\dfrac{c}{ac+2c+1}\le \dfrac 34$$

I tried bounding below the denominators by $ 2x+1$ and similarly but got only uninteresting inequalities.

Any ideas are welcome. thanks.

Answer 1

Hint: Use CS inequality to show that for positives, $$\frac{4a}{(ab+a)+(a+1)} \leqslant \frac{a}{ab+a}+\frac{a}{a+1}=\frac1{b+1}+\frac{a}{a+1}$$ Now do that for the other terms in the LHS and sum.