On Entire Solutions of Some Differential-Difference Equations
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On Entire Solutions of Some Differential-Difference Equations Kai Liu · Lianzhong Yang
Received: 16 April 2013 / Revised: 17 July 2013 / Accepted: 21 July 2013 / Published online: 27 August 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract In this paper, we will investigate the properties of entire solutions with finite order of the Fermat type difference or differential-difference equations. This is continuation of a recent paper (Liu et al. in Arch. Math. 99, 147–155, 2012). In addition, we also consider the value distribution and growth of the entire solutions of linear differential-difference equation f (k) (z) = h(z) f (z + c), where h(z) is a non-zero meromorphic function, c is a non-zero constant. Our results partially answer the question given in Liu et al. (Arch. Math. 99, 147–155, 2012). Keywords
Entire solutions · Differential-difference equations · Finite order
Mathematics Subject Classification
39B32 · 34M05 · 30D35
Communicated by Ilpo Laine. This work was partially supported by the NSFC (No. 11301260, 11101201, 11171013), the NSF of Shandong (No. ZR2010AM030), the NSF of Jiangxi (No. 20132BAB211003) and the YFED of Jiangxi (No. GJJ13078) of China. K. Liu (B) Department of Mathematics, Nanchang University, Jiangxi 330031, Nanchang, People’s Republic of China e-mail: [email protected] L. Yang School of Mathematics, Shandong University, Jinan 250100, Shandong, People’s Republic of China e-mail: [email protected]
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K. Liu, L. Yang
1 Introduction As we all know, the complex oscillation theory of meromorphic solutions of differential equations is an important topic in complex analysis. Some results can be found in [10], where Nevanlinna theory is an effective research tool. Recently, many results on meromorphic solutions of complex difference equations were rapidly obtained, such as [2–5,21] and so on. This paper is devoted to considering the entire solutions of complex differential-difference equations of certain forms. Our additional aim is to understand the similarities and differences among complex differential equations, complex difference equations, complex differential-difference equations. Nevanlinna theory will play an important role in this paper, we assume that the reader is familiar with standard symbols and fundamental results of Nevanlinna Theory [10,20]. Recall that α(z) ≡ 0, ∞ is a small function with respect to f (z), if T (r, α) = S(r, f ), where S(r, f ) is used to denote any quantity satisfying S(r, f ) = o(T (r, f )), and r → ∞ outside of a possible exceptional set of finite logarithmic measure. If f (z) is a meromorphic function, let ρ( f ) be the order of growth, λ( f ) be the exponent of convergence of the zeros. This paper is organized as follows. In Sect. 2, we will continue to consider the entire solutions of difference equations of Fermat types and improve some results in [11,12]. The properties of entire solutions of non-linear differential-difference equations of different types will be considered in Sect. 3. In Sect. 4, we will obtain some results on the
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