 Solution

A conformal map preserves oriented angles between regular smooth curves through each point. For a holomorphic map with nonzero derivative, the derivative also describes the local scale and rotation.

Indeed, if two curves ${\gamma }_1,{\gamma }_2$ meet at $z$ and have nonzero tangent vectors $v_1,v_2$, the real chain rule and complex differentiability giveBecause $f^{\prime }(z)\ne 0$, the image tangents are nonzero andTaking arguments proves preservation of oriented angles. Multiplication by $f^{\prime }(z)$ rotates every tangent by $\arg f^{\prime }(z)$ and scales every length by $|f^{\prime }(z)|$. Thus the hypothesis gives conformality at every point of $U$; global injectivity is not needed for this local conclusion.