$(i)$. Locate the maximum and minimum values of $f(z)=\ln \left|z \right| = \frac{1}{2} \ln(x^2+y^2)$ on $\overline{\Omega} = \{z \in \mathbb{R}^2: 1 \leqslant \left|z \right| \leqslant 2\}$. $(ii)$. Do the extreme values occur on the boundary?

My attempt: $(i)$. I first tried parameterizing the annulus $\overline{\Omega}$. Let $C_1$ denote the circle boundary of $\left|z \right|=2$ and $C_2$ the circle boundary of $\left|z \right|=1$. Parameterize $C_1$ counterclockwise by $x:[0,2 \pi] \to C_1$, $t \mapsto (2 \cos t, 2 \sin t)$ and parameterize $C_2$ counterclockwise by $y:[0, 2 \pi] \to C_2$, $t \mapsto (\cos t, \sin t)$.

I wanted to use the parameterized annulus to turn $f(z)$ into a one-variable function and then just take the derivatives $f^{\prime}(x(t))$, $f^{\prime}(y(t))$ and set them equal to $0$ (to find the extreme values), but I don't think they give me much information, because $f(x(t))= \frac{1}{2} \ln(2)$ and $f(y(t))=\frac{1}{2} \ln(1)$ are both constants and thus equal $0$ after differentiating?

Question $1$: Can we figure out the maximum and minimum values using this method, or do we need to try something else? (since parameterizations of circles in a plane involve cosines and sines but our function has the $x^2+y^2$ in it)

Question $2$: Is this a correct parameterization of the annulus?

My work for $(ii)$: We are given a property in the book, called the Weak Maximum Principle, which states that If $\Omega$ is a bounded domain, with $u$ continuous on $\overline{\Omega} = \Omega \cup \partial \Omega$ (union of domain and its boundary) and harmonic in $\Omega$, then $u$ is either constant on $\overline{\Omega}$ or assumes its maximum and minimum values on the boundary.

Question $3$: The function $f(z)$ is harmonic in the interior of the annulus, but how would we use this information to show that it is continuous on the boundary $\partial \Omega$ of the annulus? We need to show that it is continuous on $\Omega \cup \partial \Omega$, but I think I only know how to show that it is continuous on $\Omega$:

I know that harmonic implies that $f(z)$ is continuous on the interior, since $f(z) \in \mathcal{C}^2(\Omega) \subseteq \mathcal{C}(\Omega)$, but I don't know how to show it is also continuous on the boundary. Clearly the annulus is bounded, so using the Weak Maximum Principle, I'm pretty sure that that extrema occur on the boundary $\partial \Omega$.

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    $\begingroup$ Is this a trick question? $r \mapsto \log r$ is monotone... $\endgroup$ – Unit Apr 13 '16 at 2:26
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    $\begingroup$ The logarithm function for real argument increases monotonically. Therefore, its minimum is at $x^2+y^2=1$ and its maximum is at $x^2+y^2=2$. $\endgroup$ – Mark Viola Apr 13 '16 at 2:29
  • $\begingroup$ @Dr. MV I can see intuitively why this is the case, but how would we show that monotonically increasing implies that maximum and minimum occur on the boundary circles? $\endgroup$ – user167857 Apr 13 '16 at 3:35
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    $\begingroup$ Well, it is a real function of $x^2+y^2$. When does $x^2+y^2$ attain its maximum and minimum values inside the annulus? $\endgroup$ – Mark Viola Apr 13 '16 at 3:37
  • $\begingroup$ @Dr. MV I think I see now, thank you. $\endgroup$ – user167857 Apr 13 '16 at 4:33

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