If z is a non-real complex number, then the minimum value of Imz^5(Imz)^5 is:

Question

If $z$ is a non-real complex number, then the minimum value of $\dfrac{\operatorname{Im}z^5}{(\operatorname{Im}z)^5}$ is:

Accepted Answer

With $z=re^{i	heta}$: $\dfrac{\operatorname{Im}z^5}{(\operatorname{Im}z)^5}=\dfrac{\sin5	heta}{\sin^5	heta}=5\csc^4	heta-20\csc^2	heta+16$, a quadratic in $\csc^2	heta\ (\ge1)$, minimized (via completing the square) at value -4.

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