let $f = u +iv$ and $g = v +iu $ be nonzero analytics function on $|z| < 1$.then its follows that
choose the correct option
a) $f ' = 0 $
b) $f$ is conformal on $|z| < 1$
C) $f =cg$ for some real $c$
d) $ f$ is one -one
My attempts : i can discard option d) take $f(z) = z^2$ so it is false
for b) for f is conformal $f' \neq 0$ as take $ f$ = $e^z$ so its contradicts
im confusing about option a) and option d)
pliz help me
any hints/solution will be appreciated
thanks u
A key fact is the relationship between $f$ and $g$; namely, the analytic function $-i f$ is given by
$$-if = v - iu = \overline{g}.$$
On the other hand, if $g$ and $\overline{g}$ are both analytic, then it should be easy to see that $g$ is actually constant (via the Cauchy-Riemann equations, perhaps, or by knowing things about $z \mapsto \overline{z}$). This tells you quite a bit about $u$ and $v$, of course.
Unfortunately, this means that your work so far is incorrect. For example, if $f(z) = z^2$, then $u = x^2 - y^2$ and $v = 2xy$. However, the corresponding $g$ is not analytic. Likewise for your answer to (b).