Kyoto Journal of Mathematics

Relation between differential polynomials and small functions

Abstract

In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation

$f^{{\prime \prime }}+A_{1}(z)e^{az}f^{{\prime }}+A_{0}(z)e^{bz}f=F$,

where $a$, $b$ are complex constants and $A_{j}(z)\not\equiv 0$ $(j=0,1)$, and $F\not\equiv 0$ are entire functions such that $\max \{\rho (A_{j})\ (j=0,1),\rho (F)\}\char60 1$. We also investigate the relationship between small functions and differential polynomials $g_{f}(z)=d_{2}f^{{\prime \prime }}+d_{1}f^{{\prime }}+d_{0}f$, where $d_{0}(z),d_{1}(z),d_{2}(z)$ are entire functions that are not all equal to zero with $\rho (d_{j})\char60 1$ $(j=0,1,2)$ generated by solutions of the above equation.

Article information

Source
Kyoto J. Math., Volume 50, Number 2 (2010), 453-468.

Dates
First available in Project Euclid: 7 May 2010

https://projecteuclid.org/euclid.kjm/1273236822

Digital Object Identifier
doi:10.1215/0023608X-2009-019

Mathematical Reviews number (MathSciNet)
MR2666664

Zentralblatt MATH identifier
1203.34148

Subjects
Primary: 34M10: Oscillation, growth of solutions
Secondary: 30D35: Distribution of values, Nevanlinna theory

Citation

Belaïdi, Benharrat; El Farissi, Abdallah. Relation between differential polynomials and small functions. Kyoto J. Math. 50 (2010), no. 2, 453--468. doi:10.1215/0023608X-2009-019. https://projecteuclid.org/euclid.kjm/1273236822

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