Does this condition imply that $f'(z) \neq 0 $? Suppose $f:A \to B$ is bijective and analytic with analytic inverse. Then $f'(z) \neq 0 $ for all $z \in A $.
We know $f^{-1} \circ f (z) = z $ and so $(f^{-1})'( f^{-1} \circ f)(z) = 1 $. why from here it follows that $f'(z) \neq 0 $ ?? 

From $$ 1=\frac d{dz}z=\frac d{dz}(f^{-1}\circ f)(z)=(f^{-1})'(f(z))\cdot f'(z)$$
the factor $f'(z)$ cannot be $=0$.
