Is it true that $| \int_\gamma f|= \int_\gamma|f|$, for any closed path $\gamma$

Question

My guess is it's false. I try to give counter examples, I tried $f=z$, though $|f|$ is not holomorphic the values are same. Is the result true?

Answer 1

In general, $$\left| \int_{\gamma} f \right| \le \int_\gamma |f| |dz|.$$ For a concrete example, let $\gamma$ be the unit circle in $\mathbb{C}$ and $f(z) = z.$ The left hand side is $0$ but the right hand side is $2 \pi$.

As @Conrad points out, if you mean $\int |f| dz$ then you can take $f(z) = 1/z$. By the residue theorem $\int_{\gamma} f dz = 2\pi i $ whereas $\int_\gamma |f| dz = 0.$