About Taylor coefficients of Hardy class functions

Abstract

Let $D = \{ z \in \mathbb{C}: |z| < 1 \}$, $A(D)$ is the set of regular in $D$ functions $f$ with Taylor series $f(z)=\sum_{k=0}^{\infty} {c_k} {z^k}$, and $H^{1}$ is a class of functions $f\in A(D)$ such, that $\lim_{r \to 1} \int_{0}^{2\pi} |f(re^{i\theta})|^p d\theta<\infty.$ This class is called the Hardy class. It is known (Privalov, 1950), that in order for $f\in H^{2}$ it is necessary and sufficient that the series $\sum_ {k=0}^{\infty} |{c_k}|^2,$ and if the series $\sum_{k=0}^{\infty} |c_k|^ \frac{p}{p-1}$, for $p\geqslant2,$ then the function $f(z)=\sum_{k=0}^{\infty} {c_k} {z^k}$ will belong to the class $H^p$, $p\geqslant2$. In addition, if $f(z)\in H^{p}$, $1