On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

M. Saidani ; B. Belaïdi

Ufa mathematical journal, Tome 10 (2018) no. 1, pp. 115-134

Résumé

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].

Détail

Keywords: Order of growth, hyper-order, exponent of convergence of zero sequence, differential equation, meromorphic function.