I am trying to prove the following inequality from Analytic Number Theory which arises in the study of partial sums of a Dirichlet series:
$|\frac{1}{2\pi i}\int_{b-iT}^{b+iT}\frac{a^{s}}{s}ds|\leq \frac{8}{7}a^{b}$ where $a>0, b>1, T>0$.
My attempt:
$|\frac{1}{2\pi i}\int_{b-iT}^{b+iT}\frac{a^{s}}{s}ds|\leq \frac{1}{2\pi}\int_{-T}^{T}\frac{a^{b}dt}{\sqrt{b^{2}+t^{2}}}\leq \frac{a^{b}}{\pi}\int_{0}^{T}\frac{dt}{\sqrt{1+t^{2}}}$,
There is a hint for this problem on applying the second mean value theorem, I tried to solve this problem by this theorem, but I can not reach the desired upper bound. I also applied the rsidue theorem for contour with vertices $b+iT, iT, -iT, b-iT$ which doesn't contain the pole $s=0$ as an interior point, but I couldn't reach that upper bound.
Please help to solve this problem. Thanks.