Which one of the following statement holds for every analytics functions $f : T \rightarrow \mathbb{C}$

Question

Let $T$ be the closed units disk and dT be the unit circle. Then which one of the following statement holds for every analytics functions $f : T \rightarrow \mathbb{C}$

a) $|f|$ attains its minimum and its maximum on dT

b) $|f|$ attains its minimum on dT but need not attains its maximum on dT

c) $|f|$ attains its maximum on dT but need not attains its minimum on dT

d) $ |f|$ need not attains its maximum on dT and also it need not attains its minimum on dT

My attempts : By maximum modulus Theorem...option C will be True

Is it correct ??

Answer 1

Yes you are right.

Let me give you a different approach. You can reject the other choices by taking say $f(z)=z$, i.e. identity function.

$|f(z)|$ has maximum value $1$ which it attains on the boundary and has minimum value zero which it attains on the center of the disk.

So $a,b,d$ are false.