let $f$ be an entire function on C.Let $ g(z)=\overline f(\overline z)$ .Which of the following statements are correct?
(A) If $f(z) \in \mathbb R $ for all $z\in \mathbb R$ then $f=g$
(B) If $f(z) \in \mathbb R $ for all $z\in \{ z:Imz=0 \}\cup\{ z:Imz=a \}$ for some $a>0$,then $f(z+ia)=f(z-ia)$ for all $z\in \mathbb C$
(C)If $f(z) \in \mathbb R $ for all $z\in \{ z:Imz=0 \}\cup\{ z:Imz=a \}$ for some $a>0$,then $f(z+2ia)=f(z)$ for all $z\in \mathbb C$
(D)If $f(z) \in \mathbb R $ for all $z\in \{ z:Imz=0 \}\cup\{ z:Imz=a \}$ for some $a>0$,then $f(z+ia)=f(z)$ for all $z\in \mathbb C$
What I have done is to consider z=x+iy and $f(z)=u(x,y)+iv(x,y)$. Now (A) is clearly not true .For (B) I know $v(x,0)=v(x,a)=0$ what can we say about $v(x,-a), u(x,-a)$ when u and v both are differentiable?
I don't know how to proceed any further. Please help.I know bilinear map and cross ratio but i don't know how to use them here.Thanks.