We consider a free interpolation problem in Nevanlinna and Smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions. We also consider the related problem of characterizing positive functions in the disk having a harmonic majorant. An answer is given in terms of a dual relation which involves positive measures in the disk with bounded Poisson ARTICLE IN PRESS $ (A. Hartmann), xavier@mat.ub.es (X. Massaneda), artur@mat.uab.es (A. Nicolau), pthomas@cict.fr (P. Thomas).

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Schwarz's lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances. In this paper, inner functions which decrease hyperbolic distances as much as possible, when one approaches the unit circle, are constructed. Actually, it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function. This is related to a result of L. Carleson on the existence of singular symmetric measures. As a consequence, some results on composition operators are obtained, bringing out the importance of the Bloch spaces in this connection. Another consequence is a uniform way of producing singular measures which are simultaneously symmetric and Kahane. 1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.

The aim of this paper is to consider certain conditions on the coefficient A of the differential equation f ′′ + Af = 0 in the unit disc, which place all normal solutions f to the union of Hardy spaces or result in the zero-sequence of each non-trivial solution to be uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.

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