Abstract. This research concerns coefficient conditions for linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the second order case slowly growing solutions in H ∞ , BMOA and the Bloch space are discussed. A counterpart of the Hardy-Stein-Spencer formula for higher derivatives is proved, and then applied to study solutions in the Hardy spaces.

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.

In 1988, S. Bank showed that if {z n } is a sparse sequence in the complex plane, with convergence exponent zero, then there exists a transcendental entire A(z) of order zero such that f + A(z)f = 0 possesses a solution having {z n } as its zeros. Further, Bank constructed an example of a zero sequence {z n } violating the sparseness condition, in which case the corresponding coefficient A(z) is of infinite order. In 1997, A. Sauer introduced a condition for the density of the points in the zero sequence {z n } of finite convergence exponent such that the corresponding coefficient A(z) is of finite order.In 2010, the second author proposed a unit disc analog of Bank's first result. In the analog, {z n } is a sparse Blaschke sequence and A(z) belongs to the Korenblum space. The aim of the present paper is to introduce unit disc analogs of the two remaining results due to Bank and Sauer.

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