In complex analysis, there is a result named Lewy's theorem, which states that:
If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin $(0,0)\in \mathbb{R^2}$, then the Jacobian $J(x,y)=\left[\frac{\partial(u_1,u_2)}{\partial(x,y)}\right]$ does not vanish at the origin.
My question is: is the converse true? That is, if the Jacobian of a harmonic function $u$ does not vanish at a point , say, origin. Then is $u$ always one-one in a neighborhood of the origin?
Is a particular case of the Inverse function theorem. Being harmonic is irrelevant except because harmonic $\implies$ $C^\infty$.