Proving $1/(a+b) + 1/(b+c) + 1/(c+a) > 3/(a+b+c)$ for positive $a, b, c\,$? I have to prove that:

$$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} > \frac{3}{a+b+c},$$
where $a, b , c$ are positive real numbers.

I am thinking about using arithmetical and geometrical averages:
$$A_{3} = \frac{a_{1}+a_{2}+a_{3}}{3},$$
$$G_{3} = \sqrt[3]{a_{1}×a_{2}×a_{3}},$$
$$A_{3}\ge G_{3}.$$
However, I am not sure how to do this.
I have tried to substract one side from the other:
$$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} - \frac{3}{a+b+c} > 0.$$
I have also tried to reverse both sides of inequality.
I would appreciate if you could just give me a hint.
 A: Since $a > 0$, $b > 0$, and $c > 0$, we have
$$ 0 < a +b < a+b + c, $$
and therefore upon dividing both sides of this inequality by $(a+b)(a+b+c) > 0$, we get
$$
0 < \frac{1}{a+b+c} < \frac{1}{a+b},
$$
which implies
$$
\frac{1}{a+b} > \frac{1}{a+b+c}.  \tag{1}
$$
Similarly, we have the inequalities
$$
\frac{1}{b+c} > \frac{1}{a+b+c}  \tag{2}
$$
and
$$
\frac{1}{c+a} > \frac{1}{a+b+c}.  \tag{3}
$$
Using (1), (2), and (3), we obtain
\begin{align}
\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a } &> \frac{1}{a + b + c} + \frac{1}{a + b + c} + \frac{1}{a + b + c} \\
&= \frac{3}{a + b + c},
\end{align}
as required.
A: By AM-HM inequality
$$\frac13 \left(\frac1{a+b}+\frac1{b+c}+\frac1{c+a}\right)\ge
\frac{3}{(a+b)+(b+c)+(c+a)}.$$
Hence $\displaystyle \frac1{a+b}+\frac1{b+c}+\frac1{c+a}\ge \frac{9}2\frac1{a+b+c}$. Because $\dfrac92>3$ the problem is solved.
A: Other solution is Cauchy-Schwarz, that can be applied in one of its several versions.

*

*This is the standard one:

$[(\frac{1}{\sqrt{a+b}})^2+(\frac{1}{\sqrt{b+c}})^2+(\frac{1}{\sqrt{c+a}})^2][((\sqrt{a+b})^2+(\sqrt{b+c})^2+(\sqrt{c+a})^2]\ge 3^2$
From here the factor $9/2$, as already found, follows.

*

*Titu's inequality https://brilliant.org/wiki/titus-lemma/#
Here we directly have:
$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\ge\frac{(1+1+1)^2}{2(a+b+c)}$

*

*Holders form: https://brilliant.org/wiki/holders-inequality/ :

$(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a})^{1/2}((a+b)+(b+c)+(c+a))^{1/2}\ge 1^{1/2}+1^{1/2}+1^{1/2}$
I wrote this answer also for a reference of mine to the different forms of C.S.
