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Derivatives of Blaschke products. (English) Zbl 0551.30029

This paper is a study of the behavior of the derivative of an infinite Blaschke product, focusing on membership of the derivative B’ or fractional derivatives \(B^{\beta}\) in the classes \(A^{p,\alpha}\). Here \(A^{p,\alpha}\) \((0<p<\infty,\quad \alpha >-1)\) is the set of all f(z), holomorphic in \(| z| <1,\) and satisfying \[ \int^{1}_{0}\int^{2\pi}_{0}| f(re^{it})|^ p\quad (1- r)^{\alpha}\quad d\theta \quad dr<\infty. \] Some necessary and some sufficient conditions for the above mentioned membership are given, which generalize results of P. Ahern and the reviewer [Mich. Math. J. 21, 115-127 (1974; Zbl 0287.30034)] and P. Ahern [Indiana Univ. Math. J. 28, 311-347 (1979; Zbl 0415.30022)]. The conditions are given in terms of the zeros and occasionally the power series coefficients of B.
Reviewer: D.N.Clark

MSC:

30D50 Blaschke products, etc. (MSC2000)
30B10 Power series (including lacunary series) in one complex variable
30D55 \(H^p\)-classes (MSC2000)
26A33 Fractional derivatives and integrals
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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