Consider $A^2 = a_1^2 + a_2^2 + a_3^2$, $B^2 = b_1^2 + b_2^2 + b_3^2$ and $C^2 = c_1^2 + c_2^2 + c_3^2$.
Is the following inequality always valid, for $\forall a, b, c > 0$? $$AC - 4AB \le (a_1c_1 + a_2c_2 + a_3c_3) - 4(a_1b_1 + a_2b_2 + a_3b_3)$$
I am trying to prove that this inequality holds. It is the final step in a larger problem that I am working on (these are vector components). Please help.
What I have tried:
(a) Expand the LHS. This got very messy because of the square root exponent, and I didn't know how to proceed.
(b) Try various numbers for $a, b, c$ using brute-force, but I know this is not a sufficient method.
The inequality $AC-4AB \le (a_1c_1 + a_2c_2 + a_3c_3) - 4(a_1b_1 + a_2b_2 + a_3b_3)$ is not true for all positive reals $a_i$, $b_i$, and $c_i$. To observe that, consider $a_1=a_2=a_3 = 1$, $b_1=b_2=b_3=2$, $c_1=1$, $c_2=2$, and $c_3=3$. It is easy to verify that $AC-4AB \color{red}{>} (a_1c_1 + a_2c_2 + a_3c_3) - 4(a_1b_1 + a_2b_2 + a_3b_3)$ in this case.