zbMATH — the first resource for mathematics

Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. (English) Zbl 1085.30026
Let \(m(r,f)\) be the function and \(T(r,f)\) the characteristic of a meromorphic function \(f\). The authors prove that if \(f\) is a non-Nevanlinna proximity constant meromorphic function, \(c\in\mathbb{C}\), \(\delta< 1\) and \(\varepsilon> 0\), then \[ m\Biggl(r, {f(z+ c)\over f(z)}\Biggr)= o\Biggl({T(r+|c|,f)^{1+\varepsilon}\over r^\delta}\Biggr) \] for all \(r\) outside an exceptional set with finite logarithmic measure. This theorem is a difference analogue of the logarithmic derivative lemma, which is a useful tool in the study of complex solutions of nonlinear differential equations.
The paper contains also a number of results about the finite-order meromorphic solutions of large classes of nonlinear difference equations, obtained by using the above theorem.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B32 Functional equations for complex functions
Full Text: DOI arXiv
[1] Ablowitz, M.J.; Halburd, R.G.; Herbst, B., On the extension of the Painlevé property to difference equations, Nonlinearity, 13, 889-905, (2000) · Zbl 0956.39003
[2] Cherry, W.; Ye, Z., Nevanlinna’s theory of value distribution, (2001), Springer-Verlag Berlin
[3] Clunie, J., On integral and meromorphic functions, J. London math. soc., 37, 17-27, (1962) · Zbl 0104.29504
[4] Fuchs, L., Sur quelques équations différentielles linéares du second ordre, C. R. acad. sci. Paris, 141, 555-558, (1905) · JFM 36.0397.02
[5] Gambier, B., Sur LES équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta math., 33, 1-55, (1910) · JFM 40.0377.02
[6] R.G. Halburd, R.J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, preprint · Zbl 1119.39014
[7] Hayman, W.K., Meromorphic functions, (1964), Clarendon Press Oxford · Zbl 0115.06203
[8] Laine, I., Nevanlinna theory and complex differential equations, (1993), de Gruyter Berlin
[9] Malmquist, J., Sur LES fonctions á un nombre fini des branches définies par LES équations différentielles du premier ordre, Acta math., 36, 297-343, (1913) · JFM 44.0384.01
[10] Mohon’ko, A.Z., The Nevanlinna characteristics of certain meromorphic functions, Teor. funktsiıˇ funktsional. anal. i prilozhen., 14, 83-87, (1971), (in Russian)
[11] Mohon’ko, A.Z.; Mohon’ko, V.D., Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations, Sibirsk. mat. zh., 15, 1305-1322, (1974), (in Russian)
[12] Nevanlinna, R., Zur theorie der meromorphen funktionen, Acta math., 46, 1-99, (1925) · JFM 51.0254.05
[13] Painlevé, P., Sur LES équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta math., 25, 1-85, (1902) · JFM 32.0340.01
[14] Shimomura, S., Entire solutions of a polynomial difference equation, J. fac. sci. univ. Tokyo sect. IA math., 28, 253-266, (1981) · Zbl 0469.30021
[15] Valiron, G., Sur la dérivée des fonctions algébroïdes, Bull. soc. math. France, 59, 17-39, (1931) · JFM 57.0371.01
[16] Yanagihara, N., Meromorphic solutions of some difference equations, Funkcial. ekvac., 23, 309-326, (1980) · Zbl 0474.30024
[17] Yanagihara, N., Meromorphic solutions of some difference equations of the nth order, Arch. ration. mech. anal., 91, 169-192, (1985) · Zbl 0591.30022
[18] Yosida, K., A generalization of Malmquist’s theorem, J. math., 9, 253-256, (1933) · JFM 59.1099.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.