  We study the asymptotic behavior of the ratio of Pauli and Dirac
electromagnetic nucleon form factors, $F_2/F_1$, in time-like region for
different parametrizations built for the space-like region. We investigate how
fast the ratio $F_2/F_1$ approaches the asymptotic limits according to the
Phragm\`en-Lindel\"of theorem. We show that the QCD-inspired logarithmic
behavior of this ratio results in very far asymptotics, experimentally
unachievable. This is also confirmed by the normal component of the nucleon
polarization, $P_y$, in $e^++e^-\to N+\bar{N}$ (in collisions of unpolarized
leptons), which is a very interesting observable, with respect to this theorem.
Finally we observe that the 1/Q parametrization of $F_2/F_1$ contradicts this
theorem.
