Prove that $\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2} \text{ when } abc=1$
Solution 1
I don't think generally $\frac{a_1}{x_1} + \frac{a_2}{x_2} + \frac{a_3}{x_3} \geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3}$($a_i=10$, $x_i=1$ is counterexample), it should have been$$\frac{a_1^2}{x_1} + \frac{a_2^2}{x_2} + \frac{a_3^2}{x_3} \geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3}$$by CauchySchwartz inequality.
And if $x<y$ then $1/x>1/y$, therefore you should calculate the upper bound of $a^3(b+c) + b^3(a+c) + c^3(a+b)$, not the lower bound.
Also, it is true that $a+b+c \ge 3$ and you can prove it much simply with AMGM inequality: $a+b+c \ge 3\sqrt[3]{abc}=3$. However, it only proves that $\frac{9}{2(a+b+c)}\le \frac{3}{2}$, which is not what you want.
Solution 2
I think your first step gives a wrong inequality.
For the proof we can make the following thinks.
By Holder and AMGM we obtain: $$\sum_{cyc}\frac{1}{a^3(b+c)}=\sum_{cyc}\frac{b^3c^3}{b+c}\geq\frac{(ab+ac+bc)^3}{3\sum\limits_{cyc}(b+c)}=$$ $$=\frac{(ab+ac+bc)(ab+ac+bc)^2}{6(a+b+c)}\geq\frac{3(ab+ac+bc)^2}{6(a+b+c)}=\frac{(ab+ac+bc)^2}{2abc(a+b+c)}\geq\frac{3}{2},$$ where the last inequality it's just $\sum\limits_{cyc}c^2(ab)^2\geq0$.
Done!
Solution 3
We have $$\sum \frac{1}{a^3(b+c)} = \sum \frac{1}{a^2(ab+ac)} = \sum\frac{(bc)^2}{ab+ca}.$$ Replacing $ab=z,bc=x,ca=y$, we need to show $$\sum\frac{x^2}{y+z}\ge\frac32,\tag1$$ for $xyz=1$.
Now (1) is trivial from CS: $$2(x+y+z) \sum\frac{x^2}{y+z}=\sum (y+z) sum\frac{x^2}{y+z}\ge (x+y+z)^2$$ and $$x+y+z \ge 3\sqrt[3]{xyz}=3.$$
Related videos on Youtube
Jim Jimson
Updated on August 01, 2022Comments

Jim Jimson 10 months
So I have a possible proof but I'm not certain it's right:
$\text{As } \frac{a_1}{x_1} + \frac{a_2}{x_2} + \frac{a_3}{x_3} \geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3}\text{, we obtain } \frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)} \geq \frac{9}{a^3(b+c) + b^3(a+c) + c^3(a+b)}$
$\begin{align} a^3(b+c) + b^3(a+c) + c^3(a+b) & = a^2(\frac{1}{b} + \frac{1}{c}) + b^2(\frac{1}{a} + \frac{1}{c}) c^2(\frac{1}{a} + \frac{1}{b}) \\ & \geq \frac{4a^2}{b+c} + \frac{4b^2}{a+c} + \frac{4c^2}{a+b} \text{ by the same inequality as above} \\ & \geq 4\frac{(a+b+c)^2}{2(a+b+c)} \text{ (same reason)} \\ & \geq 2(a+b+c) \end{align}$
Now, we have:
$\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)} \geq \frac{9}{2(a+b+c)}$
I'm not sure if this is a logical progression to make though. Because if $a \geq b$ then $\frac{1}{a} \leq \frac{1}{b}$, so is this not true? I feel like it must be as I can get the desired result, but I'm confused about it.
However, if this is true it is then sufficient to prove that $a+b+c \geq 3$.
$a+b+c = a+b + \frac{1}{ab} \geq 2\sqrt{ab} + \frac{1}{ab}$
It can be proved with calculus that $\forall x\ge 0,f(x) = 2\sqrt{x} + \frac{1}{x} \geq 3$. Thus the inequality is true, and the proof is finished.
Can someone please explain either why my proof is true (namely what I am confused about) or why this false proof yields the right result?

Jim Jimson about 6 yearsYeah ok, that's much more simpler than mine, thank you. However, I am still confused as to whether my proof is correct or not. Would you be able to answer that question?

Michael Rozenberg about 6 years@Jim Jimson I think your first step is not correct.