On the growth of solutions of some higher order linear differential equations with entire coefficients
In this paper, we investigate the order and the hyper-order of entire solutions of the linear differential equation \begin{equation*} f^{\left( k\right) }+\left( D_{k-1}+B_{k-1}e^{b_{k-1}z}\right) f^{\left(k-1\right) }+ ... +\left( D_{1}+B_{1}e^{b_{1}z}\right) f^{\prime }+\left( D_{0}+A_{1}e^{a_{1}z...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2011-12-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1129 |
| _version_ | 1797830741915074560 |
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| author | H. Habib B. Belaidi |
| author_facet | H. Habib B. Belaidi |
| author_sort | H. Habib |
| collection | DOAJ |
| description | In this paper, we investigate the order and the hyper-order of entire solutions of the linear differential equation
\begin{equation*}
f^{\left( k\right) }+\left( D_{k-1}+B_{k-1}e^{b_{k-1}z}\right) f^{\left(k-1\right) }+ ... +\left( D_{1}+B_{1}e^{b_{1}z}\right) f^{\prime }+\left( D_{0}+A_{1}e^{a_{1}z}+A_{2}e^{a_{2}z}\right) f=0
\end{equation*}
where $A_{j}\left( z\right) $ $\left( \not\equiv 0\right) $ $(j=1,2)$, $ B_{l}\left( z\right) $ $\left( \not\equiv 0\right) $ $(l=1,...,k-1)$, $D_{m}$ $(m=0,...,k-1)$ are entire functions with $\max \{\sigma \left( A_{j}\right), \sigma \left( B_{l}\right), \sigma \left( D_{m}\right) \}<1$, $a_{1}$, $ a_{2}$, $b_{l}$ $(l=1,...,k-1)$ are complex numbers. Under some conditions, we prove that every solution $f\left( z\right) \not\equiv 0$ of the above equation is of infinite order and with hyper-order 1. |
| first_indexed | 2024-04-09T13:40:57Z |
| format | Article |
| id | doaj.art-d66e43605dfe4b73849ebcd652e08944 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:40:57Z |
| publishDate | 2011-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-d66e43605dfe4b73849ebcd652e089442023-05-09T07:53:01ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752011-12-0120119311310.14232/ejqtde.2011.1.931129On the growth of solutions of some higher order linear differential equations with entire coefficientsH. Habib0B. Belaidi1University of Mostaganem, Mostaganem, AlgeriaUniversity of Mostaganem, Mostaganem, AlgeriaIn this paper, we investigate the order and the hyper-order of entire solutions of the linear differential equation \begin{equation*} f^{\left( k\right) }+\left( D_{k-1}+B_{k-1}e^{b_{k-1}z}\right) f^{\left(k-1\right) }+ ... +\left( D_{1}+B_{1}e^{b_{1}z}\right) f^{\prime }+\left( D_{0}+A_{1}e^{a_{1}z}+A_{2}e^{a_{2}z}\right) f=0 \end{equation*} where $A_{j}\left( z\right) $ $\left( \not\equiv 0\right) $ $(j=1,2)$, $ B_{l}\left( z\right) $ $\left( \not\equiv 0\right) $ $(l=1,...,k-1)$, $D_{m}$ $(m=0,...,k-1)$ are entire functions with $\max \{\sigma \left( A_{j}\right), \sigma \left( B_{l}\right), \sigma \left( D_{m}\right) \}<1$, $a_{1}$, $ a_{2}$, $b_{l}$ $(l=1,...,k-1)$ are complex numbers. Under some conditions, we prove that every solution $f\left( z\right) \not\equiv 0$ of the above equation is of infinite order and with hyper-order 1.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1129linear differential equationsentire solutionsorder of growthhyper-order |
| spellingShingle | H. Habib B. Belaidi On the growth of solutions of some higher order linear differential equations with entire coefficients Electronic Journal of Qualitative Theory of Differential Equations linear differential equations entire solutions order of growth hyper-order |
| title | On the growth of solutions of some higher order linear differential equations with entire coefficients |
| title_full | On the growth of solutions of some higher order linear differential equations with entire coefficients |
| title_fullStr | On the growth of solutions of some higher order linear differential equations with entire coefficients |
| title_full_unstemmed | On the growth of solutions of some higher order linear differential equations with entire coefficients |
| title_short | On the growth of solutions of some higher order linear differential equations with entire coefficients |
| title_sort | on the growth of solutions of some higher order linear differential equations with entire coefficients |
| topic | linear differential equations entire solutions order of growth hyper-order |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1129 |
| work_keys_str_mv | AT hhabib onthegrowthofsolutionsofsomehigherorderlineardifferentialequationswithentirecoefficients AT bbelaidi onthegrowthofsolutionsofsomehigherorderlineardifferentialequationswithentirecoefficients |