Step in Apostol's IANT on Dirichlet Series

Question

I can't explain the following step in Apostol's IANT regarding Dirichlet Series.

Specifically, how does the magnitude of the following

$$\left | \int_a^b t^{s_0 -s -1} \right |$$

become the following?

$$\int_a^b t^{\sigma_0 -\sigma -1}$$

Answer 1

Because\begin{align}\left|\int_a^bA(t)t^{s_s-s-1}\,\mathrm dt\right|&\leqslant\int_a^b\left|A(t)t^{s_s-s-1}\right|\,\mathrm dt\\&\leqslant\int_a^bM\left|t^{s_0-s-1}\right|\,\mathrm dt\\&=M\int_a^bt^{\operatorname{Re}(s_0-s-1)}\,\mathrm dt\\&=M\int_a^bt^{\sigma_0-\sigma-1}\,\mathrm dt.\end{align}