Let $U$ be an open set and $f:U\to\mathbb{C}$ be a holomorphic function with real part $u(x,y)$ and imaginary part $v(x,y)$. Is it possible that $u(x,y)^2=1+v(x,y)^3$ for $x+iy\in U$

We have Cauchy-Riemann equations. But I don't know how to use them. Can anyone give any hint? thnx for your help.

• Sure it's possible: Let $f(z) \equiv 1.$ – zhw. Dec 6 '15 at 19:05

Why not use constants for $u$ and $v$? Set $u=3$ and $v=2.$ There exist infinitely many solutions like that, parametrized by a real $t.$
On the other hand, taking the derivative of the given equation with respect to $x$ or $y$, and substituting the Cauchy-Riemann equations, yields a system of two homogeneous linear equations in $u_x$ and $u_y.$ That system can only have a nonzero solution (i.e., $u$ itself nonconstant) if the determinant $9v^4+4u^2$ vanishes. Substituting the given equation into that yields a fourth-degree polynomial equation with constant coefficients in $v$, so $v$ must be constant and therefore $u$ as well.
Hint: Assume $U$ is connected. The hypotheses imply $f(U)$ is a subset of the curve $y=(x^2-1)^{1/3}.$ Think about the open mapping theorem.