Approximation of functions in $H^p$, $0<p\le1$, by generalized Riesz means with fractional exponents

For $H^p$ functions in the unit disc, $0<p\le1$, it is shown that the rate of approximation of the boundary function in the $L^p$ metric by the generalized Riesz means $R_\varepsilon^{l,\alpha}(f,z)$, $\varepsilon>0$, $(l+1)p>1$, $(\alpha+1)p>1$, is equivalent to the modulus of smoothness of fractional order $l$. This is a known result in the case of positive integer $l$.
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