Application of Opening Mapping theorem Let $f$ be a holomorphic function on open set $A$ such that $(Im(f(z))^3 + (Re(f(z))^4 =5.$ Could anyone advise me on how to use Open mapping theorem to prove $f$ is constant? Hints will suffice. Thank you.
 A: Assume $f$ is not constant on the domain $\Omega\subset{\mathbb C}$. Then there is a point $z_0\in \Omega$ with
$f'(z_0)\ne0$, and $f$ maps a suitable neighborhood $U$ of $z_0$ bijectively onto a full neighborhood $V$ of the point  $f(z_0)=:u_0+iv_0$. 
By assumption the point $(u_0,v_0)$ lies on the set $$\gamma:=\{u+iv\>|\>u^4+v^3=5\}\ .$$
The set $\gamma$ is a level set of the function $\phi(u,v):=u^4+v^3$. Since $\nabla\phi(u,v)=(4u^3,3v^2)$ is $\ne(0,0)$ for all points $(u,v)\in\gamma$, the implicit function theorem tells us that there is a window $$W:=[u_0-h,u_0+h]\times[v_0-h,v_0+h],\qquad h>0,$$ and a function $$\psi:\quad[u_0-h,u_0+h]\to [v_0-h,v_0+h],\qquad u\mapsto v=\psi(u),$$ 
$\bigl($or a similar function $v\mapsto u=\chi(v)\bigr)$ such that
$\psi(u_0)=v_0$ and $$\gamma\cap W=\{\bigl(u,\psi(u)\bigr)\>\bigm|\>u_0-h\leq u\leq u_0+h\bigr\}\ .$$
It follows that $V$ contains points not on $\gamma$, e.g., the points $(u_0,v_0+\delta)$ for sufficiently small $\delta>0$, contrary to assumption about $f$.
