Janne Gröhn

Mean growth and geometric zero distribution of solutions of linear differential equations

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 in the union of Hardy spaces or result in the zero-sequence of each non-trivial solution being uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.

Uniform separation, hyperbolic geodesics and zero distribution of solutions of linear differential equations

In this paper, separation always refers to the separation with respect to the pseudo-hyperbolic metric. If z, w ∈ D are two distinct points, then we define 〈z, w〉 ⊂ D to be the hyperbolic segment joining z and w. That is, 〈z, w〉 is a closed subarc of the unique hyperbolic geodesic which goes through z ∈ D and w ∈ D. The following result shows that a separated sequence of points is in fact uniformly separated if there exists a sufficiently dispersed intermediate sequence. In Section 4 we consider an application of Theorem 1 in which the existence of the intermediate sequence is natural.

Uniform separation through intermediate points

In this paper, separation always refers to the separation with respect to the pseudo-hyperbolic metric. If z, w ∈ D are two distinct points, then we define 〈z, w〉 ⊂ D to be the hyperbolic segment joining z and w. That is, 〈z, w〉 is a closed subarc of the unique hyperbolic geodesic which goes through z ∈ D and w ∈ D. The following result shows that a separated sequence of points is in fact uniformly separated if there exists a sufficiently dispersed intermediate sequence. In Section 4 we consider an application of Theorem 1 in which the existence of the intermediate sequence is natural.

On α-Polynomial Regular Functions, with Applications to Ordinary Differential Equations

This research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman–Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

Solutions of Complex Differential Equation Having Pre-given Zeros in the Unit Disc

Behavior of solutions of f ′′ +Af = 0 is discussed under the assumption that A is analytic in D and sup z∈D (1 − |z|)|A(z)| < ∞, where D is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that Λ ⊂ D is the zero-sequence of a non-trivial solution of f ′′ + Af = 0 where |A(z)|(1 − |z|) dm(z) is a Carleson measure if and only if Λ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.

Generalized Schwarzian derivatives and higher order differential equations

It is shown that the well-known connection between the second order linear differential equation h ″ + B ( z ) h = 0, with a solution base { h 1 , h 2 }, and the Schwarzian derivative of f = h 1 / h 2 , can be extended to the equation h ( k ) + B ( z ) h = 0 where k ≥ 2. This generalization depends upon an appropriate definition of the generalized Schwarzian derivative S k ( f ) of a function f which is induced by k −1 ratios of linearly independent solutions of h ( k ) + B ( z ) h = 0. The class k (Ω) of meromorphic functions f such that S k ( f ) is analytic in a given domain Ω is also completely described. It is shown that if Ω is the unit disc or the complex plane , then the order of growth of f ∈ k (Ω) is precisely determined by the growth of S k ( f ), and vice versa. Also the oscillation of solutions of h ( k ) + B ( z ) h = 0, with the analytic coefficient B in or , in terms of the exponent of convergence of solutions is briefly discussed.

Possible Intervals for - and -Orders of Solutions of Linear Differential Equations in the Unit Disc

In the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions of with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possible - and -orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals for - and -orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of - and -orders of functions in the solution bases.