If $f$ is holomorphic then integral of $f'(z)\overline{f(z)}$ on a close line is an imaginary number.
What I have done so far: $\frac{d}{dt}[|φ(t)|^2]=2Re(φ'(t)\overline{φ(t)})$, and since the real part of the function has an antiderivative in a closed line then it is equal to zero. However, I haven't found a way to prove that the integral of the imaginary part of the above function is non-zero.