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Suppose 
is holomorphic in Δ = {z:|z|<l} and (an)∈lp where 1≦p≦2. We prove that 
for k=1,2,…, and almost every θ. This result is sharp in the following sense: Let p∈[1,2] and ε(r) be a positive function defined on [0,1] such that limr→1-ε(r)=0. Then there exists a function holomorphic in Δ with (an)∈lp such that
for each k>1/p.
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