Whether such non constant entire function exists

Question

I need to tell whether

1. $\exists f$ non constant,entire, with $f(0)=e^{i\alpha},|f(z)|={1\over2}\forall z\in\partial\mathbb{D}$

False due to Maximum Modulas Principle

2. $\exists f$ non constant,entire, with $f(e^{i\alpha})=3,|f(z)|=1\forall |z|=3$

False due to Maximum Modulas Principle

3. $\exists f$ non constant,entire, with $f(0)=1, f(i)=0,|f(z)|\le 10\forall z\in\mathbb{C}$

False, due to Liouvilles

4. $\exists f$ non constant,entire, with $f(0)=4-3i,|f(z)|\le 5\forall z\in\mathbb{D}$

False, due to Liouvilles

5. $\exists f$ non constant,entire, with $f(z)=0\forall z=n\pi$

True, $f(z)=\sin z$

am i right in every case?

Answer 1

• Item 1. and 2. are correct provided that $\alpha$ is a real number.
• 3. is correct.
• 4. Careful: the assumption of boundedness is only on the unit disk and not the whole complex plane. Since the modulus of $f(0)$ is $5$, we can however conclude by the maximum principle.
• 5. It is correct.