Abstract
In the work we study an upper bound for a non - increasing non - negative function in the space $L^{p}(0,1)$ by the modulus of continuity of a variable increment $\omega_{p,\alpha,\psi}(f,\delta)$. We show that for the increment of the function of form $f(x)-f(x+hx^{\alpha}\psi(x))$ in the bound the modulus of continuity casts into the form $\omega_{p,\alpha,\psi}(f,\frac{\delta}{\delta^{\alpha}\psi(\frac{1}{\delta})})$. We also study the embedding $\tilde H_{p,\alpha,\psi}^\omega \subset L(\mu,\nu)(\mu \not= \nu)$ (a distant case). We obtained necessary and sufficient conditions for the parameters $p$, $\alpha$, $\mu$, $\nu$ and the functions $\psi$, $\omega$ for this embedding.