Abstract. Sufficient conditions for solutions ofand their derivatives to be in H ∞ ω (D) are given by limiting the growth of coefficientsconsists of those analytic functions f in a domain D for which |f (z)|ω(z) is uniformly bounded. In particular, the case where D is the unit disc is considered. The theorems obtained generalize and improve certain results in the literature. Moreover, by using one of the main results, one can give a straightforward proof of a classical result regarding the situation where the coefficients are polynomials.

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Abstract. We find a condition for the zeros of a Blaschke product B which guarantees that B ′ belongs to the Bergman space A p ω induced by a doubling weight ω, and show that this condition is also necessary if the zero-sequence of B is a finite union of separated sequences. We also give a general necessary condition for the zeros when B ′ ∈ A p ω , and offer a characterization of when the derivative of a purely atomic singular inner function belongs to A p ω .

For 1 2 ă p ă 8, 0 ă q ă 8 and a certain two-sided doubling weight ω, we characterize those inner functions Θ for whichThen we show a modified version of this result for p ě q. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space A p,p ω are given.2010 Mathematics Subject Classification. Primary: 30J05; Secondary: 30H10, 30H20 and 30H25.