Temperature of superconducting transition for very strong coupling in antiadiabatic limit of Eliashberg equations

It is shown that the famous Allen -- Dynes asymtotic limit for superconducting transition temperature in very strong coupling region $T_{c}>frac{1}{2pi}sqrt{lambda}Omega_0$ (where $lambdagg 1$ - is Eliashberg - McMillan electron - phonon coupling constant and $Omega_0$ - the characteristic frequency of phonons) in antiadiabatic limit of Eliashberg equations $Omega_0/Dgg 1$ ($Dsim E_F$ is conduction band half-width and $E_F$ is Fermi energy) is replaced by $T_c>(2pi^4)^{-1/3}(lambda DOmega_0^2)^{1/3}$, with the upper limit for $T_c$ given by $T_c<frac{2}{pi^2}lambda D$.