Question about Entire functions 

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*Let $D=B(z_0,R)$ be the open disc centered at $z_0$ with radius $R>0$ and $f$ be a non-constant entire function. Is it true that $f$ maps the boundary $\partial D$ of $D$ into the boundary $\partial f(D)$ of $f(D)$?


*Let $r>0$ and $f$ be an entire function such that $f(z+\frac{r}{n})=f(z)$ for all $z\in\Bbb C$ and all positive integer $n$. Is $f$ constant?


*Let $r>0$ and $f$ be an entire function satisfying $f(z+r)=f(z)$ for all $z\in\Bbb C$. Is $f$ constant?

For (1), I think Maximum Modulus theorem and Open Mapping theorem are to be applied. I have managed to show that interior goes to the interior. But I did not able to show that boundary goes to boundary, I think $f$ needs to be one-one to make it hold. Is that right?
For (2), the given condition implies that for any $z\in \Bbb C$, $f'(z)=0$ (using the definition and taking $h=r/n\to 0$). Hence $f$ is constant. I think this is ok.
For $(3)$, I don't know how to approach. I have tried many but those don't lead to any conclusion.
Any help is appreciated.

*

*Consider $f(z)=z^3$, $z_0=1, R=1$). Then $0\in \partial D$, and $0\in\partial f(D)$. But in $\partial f(D)$, $0$ is isolated; hence for boundary points sufficiently close to $0$, you obtain a counterexample.

*You are right. Actually, it would already be sufficient if thie condition holds for one $z\in\mathbb C$.

*What about $f(z)=\sin(\frac{2\pi z}{r})$?
