Note that for $x, y > 0$, $\max\{x, y\} \ge \sqrt{\frac{x^2+y^2}{2}}$. Also, $\sqrt{m^2+n^2} = \sqrt{2\left(|a|^2+|b|^2\right)}$, and $mn = \vert{a^2-b^2}\vert$. Note that \[\max\{|ac+b|,|a+bc|\} \cdot \sqrt{m^2+n^2} \ge \sqrt{\left(|ac+b|^2+|a+bc|^2\right)\left(|b|^2+|a|^2\right)}\]\[ \ge |abc+b^2|+|abc+a^2| \ge |a^2+abc-(b^2+abc)| = |a^2-b^2|.\] I used Cauchy and the Triangle Inequality. Rearranging gives the desired.