how to prove $g(z)$ is entire?

Question

let $f$ be a function such that $f$ is continuous on $\overline {\Bbb D}$ and analytic and real valued on on $\Bbb D$

($\Bbb D $ is the open unit disc centered at 0 )

define

$$g(z) =\begin{cases} f(z), & \text{if |z| $\le$ 1} \\ \overline {f(1/\overline {z})}, & \text{if |z| $\gt$ 1} \end{cases}$$

is $g$ entire?, how to prove, hints?

Answer 1

This is not true. Take $f(z)=z$. Then $g$ is not even continuous at $i$, because $g(i)=i$, but if $\lambda\in(1,+\infty)$, then$$g(\lambda i)=\overline{1/\overline{\lambda i}}=-\frac i\lambda.$$