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Several new inequalities for moduli of smoothness and errors of the bestapproximation of a function and its derivatives in the spaces $L_p$, $0<p<1$,are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in\mathbb{N}$ one has $  \omega_{r+k}(f,\d)_p\leqC({p,k,r})\d^{r+\frac{1}{p}-1}$\int_0^\d\frac{\omega_{k}(f^{(r)},t)_p^p}{t^{2-p}}{\rmd}t$^\frac{1}{p},$ where the function $f$ is such that $f^{(r-1)}$ isabsolutely continuous. Similar inequalities are obtained for the Ditzian-Totikmoduli of smoothness and the error of the best approximation of functions bytrigonometric and algebraic polynomials and splines. As an application,positive results about simultaneous approximation of a function and itsderivatives by the mentioned approximation methods in the spaces $L_p$,$0<p<1$, are derived.

Best approximations and moduli of smoothness of functions and their derivatives in $L_p$, $0