Tolman-Bondi-Lemaitre space times One can see this reference for TBL space-times.
I would like to know how the explicit expression for the function called $G$ in equations $3.108,3.108,3.110$ in the above reference is obtained. 
Also it would be nice to see some further references about TBL space-times. 

It actually just comes from integration. Equation 3.106 from that book is
$$ \dot R^2 = \left( - \frac{\partial R}{\partial t} \right)^2 = \frac FR + f $$
rearranging the terms gives (note that $\dot R < 0$)
$$ dt = - \frac{dR}{\sqrt{\frac FR + f}} $$
Now perform the substitution $z = fR/F$, giving
$$ dt = -\frac F{f^{3/2}} \frac{dz}{\sqrt{1+\frac1z}}. $$
Integrating gives
$$ t - t_0 = \frac F{f^{3/2}} \left( \sinh^{-1}\sqrt z - \sqrt{z(z+1)}\right) $$
the rest is just algebra.
I am no expert on general relativity, but the common term of that TBL spacetime seems start with "Lemaitre-Tolman". There is a review article on arXiv[1] which might help.
[1]: Kari Enqvist (2008). Lemaitre–Tolman–Bondi model and accelerating expansion. General Relativity and Gravitation 40, 2–3, pp 451–466. DOI: 10.1007/s10714-007-0553-9.
