A question on absolute values of line integrals

Question

Let $f:U\rightarrow \mathbb{C}$ be continuous and $\gamma:[a,b]\rightarrow U$ be a smooth path where $U$ is open. Then we know that $$\int_{\gamma}^{}\ f(z)dz=\int_{\gamma}^{}\ f(\gamma(t))\gamma'(t)dt$$ What I need to know is whether the following is correct. Can we write that $$|{ \int_{\gamma}^{}\ f(z)dz}| \leq \int_{\gamma}^{}\ |f(z)|dz=\int_{[a.b]}^{}\ |f(\gamma(t))|\gamma'(t)dt$$ I just need a small verification Thanks

Answer 1

In general, $dz$, equivalently $\gamma'(t)$, is not real-valued, so

$$\int_\gamma \lvert f(z)\rvert\,dz = \int_a^b \lvert f(\gamma(t))\rvert\gamma'(t)\,dt$$

need not be real. Then the inequality doesn't make sense. What is true is

$$\left\lvert \int_\gamma f(z)\,dz\right\rvert \leqslant \int_\gamma \lvert f(z)\rvert\, \lvert dz\rvert = \int_a^b \lvert f(\gamma(t))\rvert\,\lvert \gamma'(t)\rvert\,dt.$$