On the moduli of continuity of equimeasurable functions in the classes $\varphi(L)$

In the note we establish a number of relations between the moduli of continuity of the equimeasurable functions $f(x)$ and $f^*(x)$. In particular, for $f(x)\in L_p(0,1)$, $1\le p<\infty$, we have proved the inequality $\omega_p(\delta,f)\ge\frac12\omega_p(\delta,f^*),\quad\delta\in\Bigl[0,\frac12\Bigr]$.