For functions $f(z)$ in the Hardy class $H^p(0<p<\infty) $ it is known that $||f(re^{i\theta})-f(e^{i\theta})||_{L^p}<C_p\omega(1-r,f)_p$ where $\omega(\cdot,f)_p$ is the modulus of continuity of the boundary function $f(e^{i\theta})$. Extensions of this result are received to functions belonging to certain Hardy–Orlicz spaces. Proved that some resonance theorems for BMO and ReH are similar to the well-known Landau theorem in $L^p$-space.

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• I. I. Mechnikov Odessa State University
• I. I. Mechnikov Odessa National University