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}))$
