I have a basic question in my mind and wish to consult your ideas:
Suppose $\Omega_1$ and $\Omega_2$ are regions, $f$ and $g$ are nonconstant functions defined in $\Omega_1$ and $\Omega_2$, respectively, and $f(\Omega_1) \subset \Omega_2$. Define $h=g \circ f$. What can we say about the third function if
(a) both $g$ and $f$ are analytic;
(b) both $g$ and $h$ are analytic;
(c) both $h$ and $f$ are analytic.
Here I consider all possible cases.
I think in part (a) $h$ is analytic being the composition of two differentiable functions.
Actually to my mind, analyticity of $g$ implies analyticity of $h$, am I correct ? Otherwise, I can't find counterexamples on each cases. What is your suggestion?