Possible Intervals for T- and M-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 f(k)+ak-1(z)f(k-1)+⋯+a1(z)f′+a0(z)f=0 with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of s...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/928194 |
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| author | Martin Chuaqui Janne Gröhn Janne Heittokangas Jouni Rättyä |
| author_facet | Martin Chuaqui Janne Gröhn Janne Heittokangas Jouni Rättyä |
| author_sort | Martin Chuaqui |
| collection | DOAJ |
| description | 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)+ak-1(z)f(k-1)+⋯+a1(z)f′+a0(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. |
| format | Article |
| id | doaj-art-02488ab3cedb4674a5dfa27a7e4da816 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-02488ab3cedb4674a5dfa27a7e4da8162025-02-03T05:51:08ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/928194928194Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit DiscMartin Chuaqui0Janne Gröhn1Janne Heittokangas2Jouni Rättyä3Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22 Santiago, 6904411 Macual Santiago, ChileDepartment of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, FinlandDepartment of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, FinlandDepartment of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, FinlandIn 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)+ak-1(z)f(k-1)+⋯+a1(z)f′+a0(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.http://dx.doi.org/10.1155/2011/928194 |
| spellingShingle | Martin Chuaqui Janne Gröhn Janne Heittokangas Jouni Rättyä Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc Abstract and Applied Analysis |
| title | Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc |
| title_full | Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc |
| title_fullStr | Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc |
| title_full_unstemmed | Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc |
| title_short | Possible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc |
| title_sort | possible intervals for t and m orders of solutions of linear differential equations in the unit disc |
| url | http://dx.doi.org/10.1155/2011/928194 |
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