How find the maximum value of $|bc|$

Question:

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$

It is said this is answer is $$|bc|\le \dfrac{3\sqrt{3}}{16}$$

My idea: let $z=1$,then $$|a+b+c|\le 1$$ let $z=-1$,then $$|a-b+c|\le 1$$ let $z=0$, then $$|c|\le 1$$

let $|\theta|=1$,then we have $$|a(z/\theta)^2+b(z/\theta)+c|\le 1\Longrightarrow |az^2+b\theta z+c\theta^2|\le 1$$ and only this can't solve this problem,Thank you

• What value of a b c leads to your equality case? – Calvin Lin May 8 '14 at 5:15

You can assume that $a,b,c$ are reals and $b,c\ge0$. For $c$ this is obvious. For $b$ you can get this substituting $\theta z$ for $z$, as you wrote in the question. Finally, you can replace the polynomial by $\frac{(az^2+bz+c)+(\bar{a}z^2+bz+c)}2$.
At the point $e^{it}$ we have $$1 \ge |ae^{2it}+be^{it}+c|^2 = |a+be^{-it}+ce^{-2it}|^2 \\ = \big(a+b\cos t+c\cos 2t\big)^2 + \big(b\sin t+c\sin 2t\big)^2 \\ = \big(a+b\cos t+c\cos 2t\big)^2 + \big(b\sin t-c\sin 2t\big)^2 + 4bc \sin t \sin 2t \\ \ge 4bc \sin t \sin 2t.$$ The maximum of $\sin t \sin 2t$ is $\frac{16}{3\sqrt3}$ (at $t=\arcsin\frac{\sqrt2}{\sqrt3}$ where $\cos t=\frac1{\sqrt3}$, $\sin t=\frac{\sqrt2}{\sqrt3}$, $\cos 2t=-\frac13$ and $\sin 2t=\frac{2\sqrt2}{3}$). So, at this point we get the bound $bc\le \frac{3\sqrt3}{16}$.
To preserve equality in the extreme case, we have to set $b=\frac{\sqrt3}{2\sqrt2}$, $c=\frac{3}{4\sqrt2}$, $a=-\frac1{4\sqrt2}$. Then $$|ae^{2it}+be^{it}+c|^2 = |ae^{it}+b+ce^{-it}|^2 \\ = \big((a+c)\cos t+b\big)^2 + \big((a-c)\sin t\big)^2 \\ = (a-c)^2+b^2 +2(a+c)b\cos t+4ac\cos^2t \\ = \frac78+\frac{\sqrt3}4\cos t -\frac38\cos^2 t = 1 - \frac38\left(\cos t-\frac1{\sqrt3}\right)^2 \le 1.$$ Hence, $bc=\frac{3\sqrt3}{16}$ is possible.