Article
Ufa Mathematical Journal
Volume 10, Number 1, pp. 115-134
On The Growth of Solutions of Some Higher Order Linear Differential Equations With Meromorphic Coefficients
Belaidi B., Saidani M.
DOI:10.13108/2018-10-1-115
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n 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 \cite{19}.