Let $\mathbb{D}$ be the open unit disk centered at $0$ in $\mathbb{C}$ and $f : \mathbb{D} \rightarrow \mathbb{C}$ be an analytic function. let $f = u +iv$, where $u,v$ are the real and imaginary parts of $f$. If $f(z) = \sum_n a_n z^n$ is the power series of $f$, then $f$ is constant if
Choose the correct statements:
1) $f$ is analytic
2) $u(1/2) \ge u(z)$ for all $z \in \mathbb{D}$
3) The set {$ n\in \mathbb{N} \mid a_n = 0$ } is infinite
4) For any closed curve $\gamma$ in $\mathbb{D}$, $\int_{\gamma} \frac {f(z)}{(z-a)^2} =0$ for all a $\in$ $\mathbb{D}$ with $|a| \ge 1/2$
My attempts : option 1 is true by Cauchy-Riemann's theorem and option 4 is true by Cauchy's residue theorem as $\int_{\gamma} \frac {f(z)}{(z-a)^2} = 2\pi i f'(z)$ where $f (z) = 1$, and $f'(z) = 0$
I'm confused about option 2 and option 3
Pliz help me
Any hints/solution will be appreciated,,,
Thanks in advance ..