Non-Einsteinian gravitational Lagrangians assuring cosmological solutions without collapse

We discuss a method of determining the form of the hypothetical gravitational Lagrangianf(R) replacing the Einsteinian LagrangianR in order to avoid the singularity in cosmological solutions. Instead of supposing some form off(R) and then trying to solve the generalized Einstein equations, we treatf(R) as an unknown function while inserting the cosmological solution coinciding with Friedmann solution everywhere except for the singularity, which is replaced by a regular minimum of the scale factora (t) (a regular maximum of curvature). Then we findf(R) by numerical integration. The Lagrangians thus obtained for different cases (k=0, ± 1, and with the equation of state corresponding to pure radiation) have some common properties, among which ¦f(R)¦ < ¦R¦ (concavity), and the absence of asymptotes.