<s>[INST] strong inquality? Suppose $a,b,c$ are positive real number. Prove that
$$\dfrac{a(b+c)}{(a+b)(a+c)}+\dfrac{b(c+a)}{(b+a)(b+c)}+\dfrac{c(a+b)}{(c+a)(c+b)}\ge \dfrac{(a+b+c)^2}{2(a^2+b^2+c^2)}.$$
I 'd tried to use Cauchy-Schwarz's :
$$LHS=\sum_{\{a,b,c\}}\dfrac{(a(b+c))^2}{a(a+b)(a+c)(b+c)}\ge \dfrac{4(ab+bc+ca)^2}{(a+b+c)(a+b)(b+c)(c+a)}.$$
After that, I used $p,q,r$ tranformation and need to prove
$$\dfrac{4q^2}{p(pq-r)}\ge \dfrac{p^2}{2(p^2-2q)}$$
I wonder if it is a true inequality?
Somebody help me? Or give me some hint?
 [/INST] As mentioned, the modified inequality isn't always true.
Here's an approach to solve the original inequality:
We use the symmetric-sum notation $ (x, y, z) = \sum_{sym} a^xb^yc^z$ to simplify the algebraic manipulations.
Cross multiplying the denominator, we have
Original inequality
$\Leftrightarrow [ (2,1,0) + (1,1,1) ] \times 2 ( 2,0,0) \geq [( 2,0,0) + 2 (1,1,0)]  \times [ (2,1,0) + 1/3 (1,1,1) ] $
$ \Leftrightarrow 2(4, 1, 0 ) + 2(3, 2, 0 ) + 6( 3, 1, 1) + 2 (2,2,1)\geq  
(4,1,0) + 3 (3,2,0) + 3 (3, 1, 1) + 3 (3, 2, 0) + 5 (2, 2, 1) $
$\Leftrightarrow (4,1,0) + 3 (3, 1, 1)    \geq (3, 2, 0) + 3 (2, 2, 1) $.
The last inequality is true by Muirhead term wise.
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