Estimations in the theorem of the stability of Poisson distribution decompositions

Let $\Pi_\lambda$ be a Poisson distribution function and $F=F_1*F_2$ a distribution function such that either in the Lévy metric or in the uniform metric $\rho(F,\Pi_\lambda)\le\varepsilon$.
We show that, there exists a Poisson distribution function $\Pi_{\lambda_1}$ such that
$$ \rho(F_1,\Pi_{\lambda_1})<C(\lambda)\sqrt{\frac{\ln(-\ln\varepsilon)}{(-\ln\varepsilon)}}. $$