# Inequality with absolute value of a complex integral

#### bernardbb

I'm stuck trying to prove a step inside a lemma from Serre; given is

0<a<b
0<x

To prove:

$$|\int_{a}^{b}e^{-tx}e^{-tiy}dt|\leq\int_{a}^{b}e^{-tx}dt$$

I've tried using Cauchy-Schwartz for integrals, but this step is too big (using Mathematica, it lead to a contradiction); something simpler must do the trick.
Thanks in advance.

#### yyat

Use

$$\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$$

#### bernardbb

As far as I know, that only holds if f(x) is real, which it is not.

#### yyat

It also holds for complex-valued functions. For Riemann sums this is just the triangle inequality, and in the general case one can use an approximation argument.

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