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