On the hyper order of solutions of a class of higher order linear differential equations. (English) Zbl 1199.34468

Belaïdi, Benharrat; Abbas, Saïd

An. Științ. Univ. “Ovidius” Constanța, Ser. Mat. 16, No. 2, 15-30 (2008).

Summary: We investigate the order and the hyper order of entire solutions of the higher order linear differential equation \[ f^{(k)}+A_{k-1}(z) e^{P_{k-1} (z)} f^{(k-1)} +\dots +A_1 (z) e^{P_1 (z)} f'+A_0 (z) e^{P_0 (z)} f=0\;(k\geq 2), \] where \(P_j (z)\;(j=0,\dots ,k-1)\) are nonconstant polynomials such that deg \(P_j =n\;(j=0,\dots ,k-1)\) and \(A_j (z)\;(\not\equiv 0)\;(j=0,\dots ,k-1)\) are entire functions with \(\rho(A_j )<n\;(j=0,\dots ,k-1)\). Under some conditions, we prove that every solution \(f(z)\not\equiv 0\) of the above equation is of infinite order and \(\rho _2 (f)=n\).

MSC:

34M10	Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M03	Linear ordinary differential equations and systems in the complex domain
34M05	Entire and meromorphic solutions to ordinary differential equations in the complex domain

Keywords:

linear differential equations; entire solutions; hyper order