Asymptotic estimates of $p$-variational and $L^p$-moduli of continuity of functions of certain classes

We obtain sharp estimates for upper bound of $p$-variational or $L^p$-modulus of continuity on the classes of functions with given majorants of trigonometric best approximations or moduli of continuity. We also obtain the refined direct and inverse theorems of approximation in $p$-variational metric for conjugate function.