Janne Heittokangas

Complex Difference Equations of Malmquist Type

In a recent paper [1], Ablowitz, Halburd and Herbst applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typical example of their results tells us that if a complex difference equation y(z + 1) + y(z − 1) = R(z, y) with R(z, y) rational in both arguments admits a transcendental meromorphic solution of finite order, then degyR(z, y) ≤ 2. Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result (see Theorem 13) is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.

Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Complex linear differential equations of the form with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient a j (z) of (†) belongs to the weighted Bergman space A 1 k-j α where a > 0, for all j = 0,... k - 1, then all solutions are of order of growth at most a, measured according to the Nevanlinna characteristic. In the case when a = 0 all solutions are shown to be not only of order of growth zero, but of bounded characteristic. Conversely, if all solutions are of order of growth at most a > 0, then the coefficient aj (z) is shown to belong to A pj α for all p j ∈ (0, 1 k-j) and j = 0,..., k - 1. Analogous results, when the coefficients belong to certain weighted Hardy spaces, are obtained. The non-homogeneous equation associated to (†) is also briefly discussed.

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.

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.