Can anyone please help me with this problem. I must let you know from the beginning that it's not an easy one.

"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ , with $|u(t)|\leq c<b$. Knowing that the bandwidth of $u$ is $\nu$, i.e., $(\mathcal{F}u)(s)=0 , \forall s\in \mathbb{R} \backslash [-\nu,\nu]$, how can one approximate the bandwidth of $y(t)$ as accurate as possible?"

Formula for Fourier transform used: $(\mathcal{F}u)(s)=\int_{-\infty}^{\infty}u(t) e^{-2\pi i st}dt $.

PS: I know that literally the bandwidth of $y$ is infinite, but the spectrum (absolute value of Fourier Transform) clearly decreases very fast to infinity. So instead of the normal bandwidth, another definition of bandwidth can be used (you can use whatever definition you find suitable). The one that I propose is the FOBE bandwidth, which is defined by $\nu_\alpha\in \mathbb{R}$, such that:

$\int_{-\nu_\alpha}^{\nu_\alpha}(\mathcal{F}y)^2(s)ds \geq \alpha\int_{-\infty}^{\infty}(\mathcal{F}y)^2(s)ds = \alpha||y||_2^2$, with $\alpha$ a constant of choice close to 1 (e.g. $\alpha=0.99).$