Bound on a complex integral using polar form

Question

I'm a little bit confused and also a bit rusty on complex analysis... Here is my problem;

Consider a function $f:\mathbb{R}\to\mathbb{C}$. We can express this function in polar coordinates as: $f(x) = A(x)e^{ia(x)}$, with $A(x)\in\mathbb{R}_+$ and $a(x)\in \mathbb{R}$. My question is; does the following inequality hold? $$\left|\int_q^p A(x)e^{ia(x)}\mathrm{d}x \right|\le\left(\sup_{x\in[q,p]} A(x)\right)\cdot \left| \int_q^p e^{ia(x)}\mathrm{d}x\right|$$

I would say; yes. But I'm not familiar enough with complex integrals to be sure that this is indeed the case.

Sidenote: I define $\mathbb{R}_+$ as the set of all non-negative real numbers.

Answer 1

I found a counter example that shows that this does not holds true...

Choose $A(x)=e^{-x^2}$, $a(x)=x$ and $p=-q=10$. Then the inequality would state that $1.38...\le 1.08...$