I was thinking about the problem that says:
Let $f$ be a non-constant entire function such that $\left | f(z) \right |=1$ for every $z$ with $\left \lvert z \right \lvert =1$. Then which of the following option(s) is/are correct?
- (a) $f$ has a zero in the open unit disc,
- (b) $f$ always has a zero outside the closed unit disc,
- (c) $f$ need not have any zero,
- (d) any such $f$ has exactly one zero in the open unit disc.
If we take $f(z)=z^n$, then the given condition is satisfied and I think option (a) is correct. But I can not predict anything about the other options.
Please help. Thanks in advance for your time.