Complex numbers calculus problem

Let $a_1,a_2,a_3$ be three non-zero complex numbers such that their imaginary and real parts are non-negative. Find the minimum value for $$\frac{|a_1+a_2+a_3|}{^3\root\of{|a_1a_2a_3|}}$$ Please any help would be appreciated, I can't seem to find a way to turn it into a simple optimization problem without having to use 6 variables for my unknowns.

• Seems like AM-GM, thougj not exactly. – kingW3 Mar 26 '17 at 14:49
• Using the AM-GM gives $$\frac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}\ge3\frac{|a_1+a_2+a_3|}{|a_1|+|a_2|+|a_3|}$$And then what? – Mark Viola Mar 26 '17 at 15:33
• @Dr. MV i thought it was obvious after that part and was about to edit my answer but then on actually typing up my answer I've realized that i was mistaken its unclear whats next. Sorry – Ziad Fakhoury Mar 26 '17 at 16:22
• WLOG $|a_1+a_2+a_3|=1$..... Let $t=\min (\arg (a_1),\arg (a_2), \arg (a_3))$....WLOG $t=\arg a_1.$ We can replace each $a_j$ with $a_je^{-it}.$ So WLOG $a_1$ is real. The difficulty is now minimizing $|a_1a_2a_3|^{-1/3}.$ It can be $<3$. For example $a_1=1/3, a_2=i/3, a_3=(\sqrt 8\;-1)/3.$ Then $|a_1a_2a_3|^{-1/3}=3/(\sqrt 8\;-1)^{1/3}$.. – DanielWainfleet Mar 27 '17 at 5:45