On the hyper exponent of convergence of zeros of f ( j ) − φ of higher order linear differential equations

Xu Hong-Yan, Tu Jin, Zheng, Zheng Xiu-Min

Description

In this paper, we deal with the relationship between the small function and the derivative of solutions of higher order linear differential equations f ( k ) + A k − 1 f ( k − 1 ) + ⋯ + A 0 f = 0 ( k ≥ 2 ) , where A j ( z ) ( j = 0 , 1 , … , k − 1 ) are entire functions or meromorphic functions. The theorems of this paper improve the previous results given by Chen, Belaïdi, Liu. MSC: 34M10, 30D35.

Sujets

• Linear differential equation
• Type

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 7
• Langue: English

Extrait

Xu et al.Advances in Difference Equations2012,2012:114 http://www.advancesindifferenceequations.com/content/2012/1/114

R E S E A R C HOpen Access On the hyper exponent of convergence of (j) zeros off–ϕof higher order linear differential equations 1* 22 Hong-Yan Xu, Jin Tuand Xiu-Min Zheng

* Correspondence: [e-mail] 1 Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China Full list of author information is available at the end of the article

Abstract In this paper, we deal with the relationship between the small function and the derivative of solutions of higher order linear differential equations

(k) (k–1) f+A k–1 f+∙ ∙ ∙+A 0 f= 0(k≥2),

whereA j (z) (j= 0, 1,. . .,k– 1) are entire functions or meromorphic functions. The theorems of this paper improve the previous results given by Chen, Belaïdi, Liu. MSC:34M10; 30D35 Keywords:linear differential equation; hyper order; type; small function

1 Introductionand main results Complex oscillation theory of solutions of linear differential equations in the complex planeCwas started by Bank and Laine [, ]. After their well-known work, many impor- tant results have been obtained on the complex oscillation theory of solutions of linear differential equations inC, refer to [, ]. To state those correlated results, we require to give some explanation as follows. We shall assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions (see [, , ]). In addition, we will use the notationσ(f) to denote the order of meromorphic functionf(z),λ(f) to denote the exponent of convergence of the zero-sequence off(z) andλ(f) to denote exponent of convergence of distinct zero-sequence of meromorphic functionf(z), andτ(f) to denote the type of an entire functionf(z) with  <σ(f) =σ< +∞, which is defined to be (see [])

logM(r,f) τ(f) =lim sup. σ r→∞ r

We useσ  (f) to denote the hyper-order off(z),σ  (f) is defined to be (see [])

log logT(r,f) σ  (f) =lim sup. r→∞ logr

©2012 Xu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.