Inside the proof of Corollary 1.4.4 Roy's "Special Functions", the book claims the following inequality without any details :
$|\dfrac{B_2}{a+ib} - \int_0^{\infty} \dfrac{B_2(t-[t])}{(a+ib+t)^2} dt| \le \frac14 \int_0^{\infty} \dfrac{1}{(a+t)^2+b^2} dt.$
My attempt : most probably $\frac14$ comes from that maximum of $B_2 - B_2(t) = t - t^2$ is $\frac14$ for $0 \le t \le 1$. Also $|a+ib+t|^2=(a+t)^2+b^2$. How these concludes the mentioned inequality of the book?
Here $B_n=B_n(0)$ is Bernoulli numbers and $B_n(t)$ is Bernoulli polynomials.
HINT:
Use $$\frac{1}{a+ i b } = \int_{0}^{\infty} \frac{1}{(a+ i b+t)^2} dt$$