  Gamma Ray Bursts (GRBs) show evidence of different spectral shapes, light
curves, duration, host galaxies and they explode within a wide redshift range.
However, the most of them seems to follow very tight correlations among some
observed quantities relating to their energetic. If true, these correlations
have significant implications on burst physics, giving constraints on
theoretical models. Moreover, several suggestions have been made to use these
correlations in order to calibrate GRBs as standard candles and to constrain
the cosmological parameters. We investigate the cosmological relation between
low energy $\alpha$ index in GRBs prompt spectra and the redshift $z$. We
present a statistical analysis of the relation between the total isotropic
energy $E_{iso}$ and the peak energy $E_p$ (also known as Amati relation) in
GRBs spectra searching for possible functional biases. Possible implications on
the $E_{iso}$ vs $E_p$ relation of the $\alpha$ vs $(1+z)$ correlation are
evaluated. We used MonteCarlo simulations and the boostrap method to evaluate
how large are the effects of functional biases on the $E_{iso}$ vs $E_p$. We
show that high values of the linear correlation coefficent, up to about 0.8, in
the $E_{iso}$ vs $E_p$ relation are obtained for random generated samples of
GRBs, confirming the relevance of functional biases. Astrophysical consequences
from $E_{iso}$ vs $E_p$ relation are then to be revised after a more accurate
and possibly bias free analysis.
