  In the paper, we study inequalities for the best trigonometric approximations
and fractional moduli of smoothness involving the Weyl and Liouville-Gr\"unwald
derivatives in $L_p$, $0<p<1$. We extend known inequalities to the whole range
of parameters of smoothness as well as obtain several new inequalities. As an
application, the direct and inverse theorems of approximation theory involving
the modulus of smoothness $\omega_\beta(f^{(\alpha)},\delta)_p$, where
$f^{(\alpha)}$ is a fractional derivative of the function $f$, are derived. A
description of the class of functions with the optimal rate of decrease of a
fractional modulus of smoothness is given.
