$\varphi$ is closed curve in $\mathbb{C}$, $f: G \rightarrow \mathbb{C}$ holomoprh on a doman $G$ with image of $\varphi$ subset of G. If $(f(z): z\in \varphi^{*})$ ($\varphi^{*}$ image of $\varphi$) has NO non-positiv real Points then: $$\int_{\varphi } {f}'/f = 0$$
How to show this part? I know this is complex logarithm but I´m not sure how to use the fact.
$\log z$ has a holomorphic branch on the slit plane with the non-negative reals removed. If we call this branch $l(z)$ then $$\frac d{dz}l(f(z))=\frac{f'(z)}{f(z)}$$ and the integral becomes $$\int_{\varphi}\frac d{dz}l(f(z))\,dz=0$$ as we are integrating a derivative over a closed contour.