whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions In each of the case below, determine whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions.
($1$) $f(0)=e^{i\alpha}$ and $|f(z)|=1/2$ for all $z \in Bdr \Delta$.
($2$) $f(e^{i\alpha})=3$ and $|f(z)|=1$ for all $z$ with $|z|=3$.
($3$) $f(0)=1$ , $f(i)=0$ , and $|f(z)| \le 10$ $z \in \mathbb{C}$.
($4$) $f(0)=1 , f(i)=0$ , and $|f(z)| \le 5$ $ \forall z \in \Delta$.
($5$) $f(z) =0 $ for all $z=n \pi$ , $n\in \mathbb{Z}$  

My thought:-
($1$)No. by Maximum-modulus Theorem $|f(z)|$ has maximum value at the boundary,Then $1/2$ would be the maximum value but $|f(0)|=1$ which is a contradiction.
($2$)No . by same argument as above.   
($3$)No. by liouvilles Theorem $f(z)$ is bounded hence must be constant.   
($4$) I think it is true but not sure.   
($5$) True. $\sin z$ is the example.  
please somebody verify my answers.

Looks good to me. Surely you can find an example for (4)? A first degree polynomial should do the job.
