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Уфимский математический журнал, 2018, том 10, выпуск 1, страницы 118–136
(Mi ufa414)
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On the growth of solutions of some higher order linear differential equations with meromorphic coefficients
M. Saidani, B. Belaїdi Department of Mathematics,
Laboratory of Pure and Applied Mathematics,
University of Mostaganem (UMAB),
B. P. 227 Mostaganem-(Algeria)
Аннотация:
In this paper, by using the value distribution theory, we study the growth and the oscillation of
meromorphic solutions of the linear differential equation
\begin{align*}
f^{(k) }&+\left( A_{k-1,1}(z) e^{P_{k-1}(z) }+A_{k-1,2}(z) e^{Q_{k-1}(z) }\right)
f^{\left( k-1\right) }
\\
&
+\cdots +\left( A_{0,1}(z) e^{P_{0}(z)
}+A_{0,2}(z) e^{Q_{0}(z) }\right) f=F(z),
\end{align*}
where $A_{j,i}(z) \left( \not\equiv 0\right) $ $\left(
j=0,\ldots,k-1\right),$ $F(z) $ are meromorphic functions of a
finite order, and $P_{j}(z),Q_{j}(z) $ $
(j=0,1,\ldots,k-1;i=1,2)$ are polynomials with degree $n\geqslant 1$. Under some
conditions, we prove that as $F\equiv 0$, each meromorphic solution $f\not\equiv 0$ with poles of uniformly bounded multiplicity is of infinite order and satisfies $\rho _{2}(f)=n$ and as $F\not\equiv 0$, there exists at most one exceptional solution $f_{0}$ of a finite order, and all other transcendental meromorphic solutions $f$ with poles of uniformly bounded multiplicities satisfy ${\overline{\lambda }(f)=\lambda (f)=\rho \left( f\right) =+\infty }$ and $\overline{\lambda }_{2}\left( f\right) =\lambda _{2}\left( f\right)
=\rho _{2}\left( f\right) \leq \max \left\{ n,\rho \left( F\right) \right\}.$ Our results extend the previous results due Zhan and Xiao [19].
Ключевые слова:
Order of growth, hyper-order, exponent of convergence of zero sequence, differential equation, meromorphic function.
Поступила в редакцию: 06.01.2017
Образец цитирования:
M. Saidani, B. Belaïdi, “On the growth of solutions of some higher order linear differential equations with meromorphic coefficients”, Уфимск. матем. журн., 10:1 (2018), 118–136; Ufa Math. J., 10:1 (2018), 115–134
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/ufa414 https://www.mathnet.ru/rus/ufa/v10/i1/p118
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Страница аннотации: | 248 | PDF русской версии: | 103 | PDF английской версии: | 14 | Список литературы: | 53 |
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