Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations
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RESEARCH
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Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations Qi Jianming1,3, Li Yezhou2 and Yuan Wenjun3* * Correspondence: [email protected] 3 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, People’s Republic of China Full list of author information is available at the end of the article
Abstract In this article, by means of the normal family theory we estimate the growth order of entire solutions of some algebraic differential equations and improve the related results of Bergweiler, Barsegian, and others. We also estimate the growth order of entire solutions of a type system of a special algebraic differential equations. We give some examples to show that our results are sharp in special cases. Mathematica Subject Classification (2000): Primary 34A20; Secondary 30D35. Keywords: Meromorphic functions, Nevanlinna theory, Normal family, Growth order, Algebraic differential equation
1. Introduction and main results Let f(z) be a meromorphic function in the complex plane. We use the standard notation of the Nevanlinna theory of meromorphic functions and denotes the order of f(z) by l(f) (see [1-3]). Let ℂ be the whole complex domain. Let D be a domain in ℂ and F be a family of meromorphic functions defined in D. F is said to be normal in D, in the sense of Montel, if each sequence {fn } ⊂ F has a subsequence {fnj } which converse spherically locally uniformly in D, to a meromorphic function or ∞ (see [1]). In general, it is not easy to have an estimate on the growth of an entire or meromorphic solution of a nonlinear algebraic differential equation of the form P(z, w, w , . . . , w(k) ) = 0,
(1:1)
where P is a polynomial in each of its variables. A general result was obtained by Gol’dberg [4]. He obtained Theorem 1.1. All meromorphic solutions of algebraic differential equation (1.1) have finite order of growth, when k = 1. For a half century Bank and Kaufman [5] and Barsegian [6] gave some extensions or different proofs, but the results have not changed. Barsegian [7] and Bergweiler [8] have extended Gol’dberg’s result to certain algebraic differential equations of higher order. In 2009, Yuan et al. [9], improved their results and gave a general estimate of order of w(z), which depends on the degrees of coefficients of differential polynomial for w(z). In order to state these results, we must introduce some notations: m Î N = © 2012 Jianming et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.
Jianming et al. Advances in Difference Equations 2012, 2012:6 http://www.advancesindifferenceequations.com/content/2012/1/6
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{1, 2, 3,...}, rj Î N0 = N ∪ {0} for j = 1, 2,..., m, and put r = (r1, r2,..., rm). Define Mr[w] (z) by Mr [w](z) := [w (z)]r1 [w
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