Maximum /Minimum Modulus theorem for Harmonic Function ( Corollary 6.16 )

Question

Suppose thatt $u(x,y)$ is a real valued non constant harmonic function on a bounded domain D. Then $u(x,y)$ can not attain its maximum or minimum value in $D$.

I am studing complex analysys in $S. Ponnusammy$. It used the following result

$\bullet$ (Maximum Modulus Theorem ) Suppose $f$ is analytic function in a bounded domain $\overline D$. Then $|f(z)|$ attains its maximum at some point on the boundary $\partial D$ of $D$.

$\bullet$ ( Maximum Modulus Principal) Suppose $f$ is analytic function on a domain $D$ and a is a point in $D$ such that $|f(z)| < |f(a) |$ hold for all $ z \in D$. Then $f$ is constant.

Suppose $D$ is simply connected , then there exist analytic function $f = u + \iota v$. He apply the maximum modulus theorem to $g(z) = e^{f(z)}$.

Why he apply Maximum Modulus Theorem to $g(z)$ rather than $f(z)$. Can I use directly Maximum Modulus Principal. Please Clear my doubt. Any help would be appreciated. Thank you

Answer 1

Using the maximum principle on $f$ is of course allowed because the hypothesis are satisfied. But this is not very concluding : the only information we know is that $u^2 + v^2$ take its maximum on $\partial D$ .

Using $g$ is more appropriate : $|g| = |e^u|$ and therefore $g$ verifie the Maximum Modulus Principle $\Leftrightarrow$ $u$ verifies the MMP too.

But in fact it's true for $v$ too : take $f' = if$ and $g' = e^{f'}$. Because $g'$ is analytic, the maximum principle applies, and by the same argument (namely that $|g'| = |v|$) you can conclude that $v$ also take its maximum on $\partial D$ (until it's constant).