  We present the most extensive sample of 45 type I (short) and 275 type II
(long) gamma-ray bursts (GRB) with known redshift to investigate the
correlation between the rest frame peak energy, Ep,i and the total isotropic
equivalent energy, Eiso of the prompt emission (Amati relation). The Ep,i -
Eiso correlation for type I bursts is found to be well-distinguished from the
one constructed for type II bursts and has a similar power-law index value, a =
0.4, which possibly indicates the same emission mechanism of both GRB types. We
show that the initial pulse complex (IPC) of type I bursts with an extended
emission and regular type I bursts follow the same correlation. We obtain
similar results for type II bursts associated with Ic supernovae and for
regular type II bursts. Three possible outliers from the Ep,i - Eiso
correlation for type II subsample are detected. Significant evolution of the
Ep,i - Eiso correlation with redshift for type II bursts is not found. We
suggest the new classification method, based on the Ep,i - Eiso correlation and
introduce two parameters, EH and EHD. EHD is found to be the most reliable
parameter for the blind type I - type II classification, which can be used to
classify GRBs with no redshift.
