<s>[INST] Asymmetric inequality in three variables $\frac{3(a+b)^2(b+c)^2}{4ab^2c} \geq 7+\frac{5(a^2+2b^2+c^2)}{(a+b)(b+c)}$ Consider three real positive variables $a,\ b$ and $c$. Prove that the following inequality holds:
$$\frac{3(a+b)^2(b+c)^2}{4ab^2c} \geq 7+\frac{5(a^2+2b^2+c^2)}{(a+b)(b+c)}$$
My progress: We can prove that both sides are greater than $12$ using AM-GM:
$$LHS \geq \frac{3 \cdot 4ab \cdot 4bc}{4a^2bc} = 12$$
and 
$$RHS \geq 7+\frac{5[(a+b)^2+(b+c)^2]}{2(a+b)(b+c)} \geq 7+5 = 12$$
So, substract $12$ from both sides and write the inequality into:
$$3\cdot \frac{(a+b)^2(b+c)^2-16ab^2c}{4ab^2c} \geq 5 \cdot \frac{a^2+b^2+c^2-ab-bc-ca}{(a+b)(b+c)}$$
or
$$3\cdot \frac{(b+c)^2(a-b)^2+4ab(b-c)^2}{4ab^2c} \geq \frac{5}{2}\cdot \frac{(a-b)^2+(b-c)^2+(c-a)^2}{(a+b)(b+c)}$$
My next idea was to use $(c-a)^2\leq 2[(a-b)^2+(b-c)^2]$ and write it into a sum of square form with only $(a-b)^2$ and $(b-c)^2$. However, I couldn't reach significant progress.
 [/INST] Another proof:
Since the inequality is symmetric in $a$ and $c$, WLOG, assume that $a \ge c$.
Due to homogeneity, assume that $c = 1$.
Let $a = 1 + s$ for $s \ge 0$.
We split into two cases:
1) $0 < b \le 1$: Let $b = \frac{1}{1+t}$ for $t \ge 0$. We have
\begin{align}
\mathrm{LHS} - \mathrm{RHS} &= \frac{1}{4(2+t)(st+s+t+2)(1+s)(1+t)^2}f(s,t) 
\end{align}
where
\begin{align}
f(s,t) &= 3 s^3 t^6+27 s^3 t^5+9 s^2 t^6+79 s^3 t^4+90 s^2 t^5+9 s t^6+109 s^3 t^3\\
&\quad +281 s^2 t^4+99 s t^5+3 t^6+78 s^3 t^2+412 s^2 t^3+314 s t^4+36 t^5\\
&\quad +28 s^3 t+324 s^2 t^2+452 s t^3+112 t^4+4 s^3+140 s^2 t+304 s t^2\\
&\quad +152 t^3+28 s^2+76 s t+76 t^2.
\end{align}
Clearly, $f(s,t) \ge 0$. True.
2) $b > 1$: Let $b = 1+r$ for $r > 0$. We have
\begin{align}
\mathrm{LHS} - \mathrm{RHS} &= \frac{1}{4(1+s)(1+r)^2(2+s+r)(2+r)}g(s,r)
\end{align}
where
\begin{align}
g(s, r) &= 3 r^6+9 r^5 s+9 r^4 s^2+36 r^5+22 r^4 s+44 r^3 s^2\\
&\quad +112 r^4+4 r^3 s+44 r^2 s^2+76 r^3+9 r s^2+9 s^2+\tfrac{15}{16} s^3\\
&\quad + 19 (2 r-s)^2+19 r (2 r-s)^2+3 (r-1)^2 r s^3+4 s^3 (r-\tfrac78)^2.
\end{align}
Clearly, $g(s,r) > 0$. True.
We are done.
</s>