MAT334-2018F > MAT334--Lectures & Home Assignments

3.2 Q7

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ashley:
I have no ideas how to start this question. Can someone help me to solve? Thank you.

Kris:
The function $h(z)=e^{F(z)}$ is analytic on the disc $\{\mid$$z-z_0\mid$$\leq$$r\}, it never equals zero, and \mid$$h(z_0)\mid=1$.
Hence the maximum and the minimum are attained on the boundary circle $\{\mid$$z-z_0\mid$$=r\}$.
Let $z_{max}$, $z_{min}$ be the corresponding points, so $1<\mid$$h(z_{max})\mid=e^{Re(z_{max})}, 1>\mid$$h(z_{min})\mid$=$e^{Re(z_{min})}$.
We deduce that Re($f(z_{max})$)$>0>Re(f(z_{min}))$