On difference equations concerning Schwarzian equation

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• 作者：Shuang-Ting Lan ; Zong-Xuan Chen
• 关键词：meromorphic solution ; difference ; Nevanlinna exceptional value
• 刊名：Advances in Difference Equations
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1585KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847
• 卷排序：2017

文摘

Consider the difference equation $$\biggl[\frac{\Delta^{3}f(z)}{\Delta f(z)}-\frac{3}{2} \biggl(\frac {\Delta^{2}f(z)}{\Delta f(z)} \biggr)^{2} \biggr]^{k} =\frac{P(z,f(z))}{Q(z,f(z))}, $$ where \(P(z,f)\) and \(Q(z,f)\) are prime polynomials in \(f(z)\) with \(\deg_{f}P=p, \deg_{f}Q=q\), and \(d=\max\{p,q\}>0\). We give the supremum of d, an estimation of the sum of Nevanlinna exceptional values of meromorphic solution \(f(z)\) of the equation, and study the value distributions of their difference \(\Delta f(z)\) and divided difference \(\frac{\Delta f(z)}{f(z)}\).Keywordsmeromorphic solutiondifferenceNevanlinna exceptional valueMSC30D3534A201 Introduction and main resultsIn this paper, we use the basic notions of Nevanlinna theory, such as \(T(r,f)\), \(m(r,f)\), \(N(r,f)\), and so on; see [1–3]. Let \(S(r,f)\) denote any quantity satisfying \(S(r,f) = o (T(r,f) )\) for all r outside a set of finite logarithmic measure. We call fan admissible solution of a difference (or differential) equation if all coefficients α of the equation satisfy \(T(r,\alpha)=S(r,f)\). In addition, we denote by \(\sigma(f)\) the order of growth of a meromorphic function \(f(z)\) and by \(\lambda(f)\) and \(\lambda(\frac{1}{f} )\), respectively, the exponents of convergence of zeros and poles of \(f(z)\), which are defined by $$\sigma(f)=\mathop{\overline{\lim}}_{r\rightarrow\infty}\frac{\log T(r,f)}{\log r},\qquad\lambda(f)= \mathop{\overline{\lim}}_{r\rightarrow\infty}\frac{\log N (r,\frac {1}{f} )}{\log r}, \qquad\lambda\biggl( \frac{1}{f} \biggr)=\mathop{\overline{\lim}}_{r\rightarrow\infty}\frac{\log N(r,f)}{\log r}. $$ If \(\lambda(f-a)<\sigma(f)\), then a is called a Borel exceptional value of f.For \(a\in \mathbb {C}\cup\{\infty\}\), we denote by \(\delta(a, f)\) the deficiency of a to \(f(z)\), which is defined by $$\delta(a,f)=\mathop{\underline{\lim}}_{r\rightarrow\infty}\frac{m (r,\frac {1}{f-a} )}{T(r,f)}\quad(\mbox{if } a \in \mathbb {C}),\qquad\delta(\infty, f)=\mathop{\underline{\lim}}_{r\rightarrow\infty} \frac{m(r,f)}{T(r,f)}. $$ Obviously, \(\delta(a,f)\geq0\). If \(\delta(a,f)>0\), then a is called a Nevanlinna exceptional value of f. The forward differences \(\Delta ^{n}f(z), n\in \mathbb {N}^{+}\), are defined in the standard way [4] by $$ \Delta f(z)=f(z+1)-f(z), \qquad\Delta^{n+1} f(z)=\Delta^{n}f(z+1)- \Delta^{n}f(z). $$Ishizaki [5] studied Schwarzian differential equations and obtained the following: