Risto Korhonen

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.

Holomorphic curves with shift-invariant hyperplane preimages

If f : C ! P n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation �(z) = z +c, then f is periodic with period c 2 C. This result, which can be described as a difference analogue of M. Greens Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartans second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.

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.

A Difference Picard Theorem for Meromorphic Functions of Several Variables

It is shown that if n ∈ ℕ, c ∈ ℂn, and three distinct values of a meromorphic function f: ℂn sr 1 of hyper-order gV(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂn sr ℂn, τ (z) = z + c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green’s Picard-Type Theorem in the special case where gV(f) < 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the Lemma on the Logarithmic Derivative and of the Second Main Theorem of Nevanlinna theory for meromorphic functions ℂn → ℙ P1 are given, and their applications to partial difference equations are discussed.

Order reduction method for linear difference equations

An order reduction method for homogeneous linear difference equations, analogous to the standard order reduction of linear differential equations, is introduced, and this method is applied to study the Nevanlinna growth relations between meromorphic coefficients and solutions of linear difference equations.

NONDECREASING FUNCTIONS, EXCEPTIONAL SETS AND GENERALIZED BOREL LEMMAS

According to the classical Borel lemma, any positive nondecreasing continuous function T satisfies T ( r +1/ T ( r ))≤2 T ( r ) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.

Intersections and Unions of Weighted Bergman Spaces

Certain intersection and unions with respect to different parameters of the weighted Bergman spaces are shown to be equal. These results are extended to a more general class of function spaces. Some of the results proved here play an important role in the study of complex linear differential equations in the unit disc.