Growth and complex oscillation of differential polynomials generated by solutions of differential equations. (English) Zbl 1263.34125

Belaïdi, Benharrat; El Farissi, Abdallah

Int. J. Qual. Theory Differ. Equ. Appl. 4, No. 1, 77-87 (2010).

Summary: In this paper, we investigate the fixed points and the hyper order of the differential polynomial \(g_f=d_2f^{\prime\prime}+d_1f^\prime+d_0f\), where \(d_0(z),d_1(z),d_2(z)\) are entire functions that are not all equal to zero with \(p(d_j)<\infty(j=0,1,2)\) generated by solutions of the differential equation \[ f^{\prime\prime}+A_1(z)f^\prime+A_0(z)f=F, \] where \(A_1(z),\, A_0(z)\, (\not\equiv 0),\,F\) are entire functions of finite order. Because of the control pf differential equation, we can obtain some precise estimates of their hyper order and fixed points. We also investigate the relation between infinite order solutions of higher order linear differential equations with entire coefficients and finite order entire functions.

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

Keywords:

differential polynomials; linear differential equations; entire solutions; hyper order; exponent of convergence of the sequence of distinct zeros; hyper exponent of convergence of the sequence of distinct zeros