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. We prove the estimate C(p)\le\pi e^p/[p(1-p)] in the Hardy-Littlewood inequality /BR \sum_{n=0}^\infty \frac{|a_n(f)|^p}{(n+1)^{2-p}}\le C(p)(\| f\|_p)^p. /BR We also give explicit estimates for sums \sum|a_n(f)\lambda_n|^s in the mixed norm space H(1,s,\beta). In this way we obtain a new version of some results by Blasco and by Jevti\u{c} and Pavlovi\u{c}.

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