Inequalities in approximation theory involving fractional smoothness in L_p, 0<p<1

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.