  The properties of the square bias transformation are studied, in particular,
the precise moment-type estimate for the $L_1$-metric between the transformed
and the original distributions is proved, a relation between their
characteristic functions is found. As a corollary, some new moment-type
estimates for the proximity of arbitrary characteristic function with zero mean
and finite third moment to the normal one with zero mean and the same variance
are proved involving the double integrals of the square- and zero- bias
transformations.
