On the coefficient multipliers theorem of Hardy and Littlewood

Let $a_n(f)$ be the Taylor coefficients of a holomorphic function $f$ which belongs to the Hardy space $H^p$, $0<p<1$. We prove the estimate $C(p)\leq\pi\epsilon^p/[p(1-p)]$ in the Hardy-Littlewood inequality
$$ \sum_{n=0}^\infty\frac{|a_n(f)|^p}{(n+1)^{2-p}}\leq C(p)(\| f \|_p)^p. $$
We also give explicit estimates for sums $\sum|a_n(f)\lambda_n|^s$ the mixed norm space $H(1,s,\beta)$. In this way we obtain a new version of some results by Blasco and by Jevtič and Pavlovič.