Abstract and Applied Analysis
Volume 2011 (2011), Article ID 928194, 25 pages
Research Article

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

1Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22 Santiago, 6904411 Macual Santiago, Chile
2Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 19 May 2011; Accepted 5 July 2011

Academic Editor: Jean Michel Combes

Copyright © 2011 Martin Chuaqui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 𝑓 ( 𝑘 ) + 𝑎 𝑘 1 ( 𝑧 ) 𝑓 ( 𝑘 1 ) + + 𝑎 1 ( 𝑧 ) 𝑓 + 𝑎 0 ( 𝑧 ) 𝑓 = 0 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 𝑇 - and 𝑀 -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 𝑇 - and 𝑀 -orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of 𝑇 - and 𝑀 -orders of functions in the solution bases.