Inequality $a^2+2b^2+8c^2\geq2a(b+2c)$

Question

Can someone prove this inequality for the real numbers a,b,c?

$$a^2+2b^2+8c^2\geq2a(b+2c)$$

I have tried simple manipulation of the terms to get quadratic expressions, but since one cannot factor the $4ac$ on the right side, I abandoned that approach. Then I tried turning it into an expression where I could apply AM-GM, but that did not work either. Maybe something like Muirhead, though I do not know where to start with that? Any and all help would be greatly appreciated!!

Answer 1

Your inequation is equivalent to

$$(a-b-2c)^2+(b-2c)^2 \ge 0 \tag{1}$$

The advantage of expression (1) is that it allows to clearly obtain the limit cases where there is an equality sign in (1) instead of a $">"$ symbol, i.e., iff

$$a=2b=4c$$

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Edit: One can wonder how I have found expression (1). If you happen to know the concept of matrix associated with a quadratic form, here is the explanation:

$$a^2+2b^2+8c^2-2a(b+2c)$$

$$=\begin{pmatrix}a&b&c\end{pmatrix}\begin{pmatrix} 1& -1&-2\\ -1&2&0\\ -2&0&8 \end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}$$

which can be transformed, using the so-called (incomplete) Cholesky factorization:

$$=\begin{pmatrix}a&b&c\end{pmatrix}\begin{pmatrix} 1&0&0\\ -1&1& \ \ 0\\ -2&-2 & \ \ 0 \end{pmatrix}\begin{pmatrix} 1&-1&-2\\ \ \ 0&\ \ 1&-2\\ 0&0 &0 \end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}$$

$$=\begin{pmatrix}(a-b-2c)&(b-2c)&0\end{pmatrix}\begin{pmatrix}(a-b-2c)\\(b-2c)\\0\end{pmatrix}$$

$$=(a-b-2c)^2+(b-2c)^2$$

Answer 2

Hints:

$$\frac12a^2 - 2ab + 2b^2 = \frac12(a-2b)^2 \ge 0$$

$$\frac 12 a^2 - 4ac + 8c^2 = \frac12(a-4c)^2 \ge 0$$

Answer 3

$a^2 - a(2b+4c) + 2b^2+8c^2 \ge 0$ since $\triangle = (2b+4c)^2 -4(2b^2+8c^2) = -4b^2+16bc-16c^2 = -4(b-2c)^2 \le 0$.

Answer 4

$$a^2+2b^2+8c^2=\frac{1}{2}a^2+2b^2+\frac{1}{2}a^2+8c^2$$ Now By AM-GM : $$\cases{\frac{1}{2}a^2+ 2b^2 \geq 2ab \\ \frac{1}{2}a^2+8c^2\geq4\sqrt{a^2c^2}=4ac}$$

Adding them together: $$a^2+2b^2+8c^2 \geq2ab+4ac$$

And that’s a good place to stop.

Answer 5

$a,b,c$ are real numbers. $$a^2+2b^2+8c^2\geq2a(b+2c)~~~~(1)$$ if for all real values of $a$ $$a^2-a(b+2c)^2+3b^2+8c^2\ge 0~~~~~(2)$$ $$\implies B^2 \le 4AC$$, as $A>0$, then $$(b+2c)^2 \le 4(3b^2+8c^2) \implies -11b^2+4bc-28c^2\le 0$$ $$\implies 11b^2-4bc+28c^2\ge 0~~~~~(3)$$ In this quadratic $$D=B^2-4AC=4c^2-11.28 c^2 \le 0~~~~(4)$$ So (3) holds for all real values of $b$ and (4) holds for all real values of $c$. Hence, this proves that (1), holds for all real values of $a,b,c$.