Studies of Differences from the point of view of Nevanlinna Theory

Jianhua, Zheng; Korhonen, Risto

doi:10.1090/tran/8069

Studies of Differences from the point of view of Nevanlinna Theory
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by Zheng Jianhua and Risto Korhonen PDF

Trans. Amer. Math. Soc. 373 (2020), 4285-4318 Request permission

Abstract:

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a $\delta$-subharmonic function is established allowing the case of hyper-order equal to one and minimal hyper-type, which improves the condition of the hyper-order less than one. Finally, we make a careful discussion of a well-known difference equation and give the possible forms of the equation under a growth condition for the solutions.

References

M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), no. 3, 889–905. MR 1759006, DOI 10.1088/0951-7715/13/3/321
William Cherry and Zhuan Ye, Nevanlinna’s theory of value distribution, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. The second main theorem and its error terms. MR 1831783, DOI 10.1007/978-3-662-12590-8
Yik-Man Chiang and Shao-Ji Feng, On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105–129. MR 2407244, DOI 10.1007/s11139-007-9101-1
Yik-Man Chiang and Xu-Dan Luo, Difference Nevanlinna theories with vanishing and infinite periods, Michigan Math. J. 66 (2017), no. 3, 451–480. MR 3695346, DOI 10.1307/mmj/1496995338
J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17–27. MR 143906, DOI 10.1112/jlms/s1-37.1.17
Albert Edrei and Wolfgang H. J. Fuchs, Bounds for the number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc. (3) 12 (1962), 315–344. MR 138765, DOI 10.1112/plms/s3-12.1.315
Anatoly A. Goldberg and Iossif V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii; With an appendix by Alexandre Eremenko and James K. Langley. MR 2435270, DOI 10.1090/mmono/236
B. Grammaticos, T. Tamizhmani, A. Ramani, and K. M. Tamizhmani, Growth and integrability in discrete systems, J. Phys. A 34 (2001), no. 18, 3811–3821. MR 1840846, DOI 10.1088/0305-4470/34/18/309
R. G. Halburd and R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 443–474. MR 2308234, DOI 10.1112/plms/pdl012
Rod Halburd and Risto Korhonen, Growth of meromorphic solutions of delay differential equations, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2513–2526. MR 3626508, DOI 10.1090/proc/13559
R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477–487. MR 2185244, DOI 10.1016/j.jmaa.2005.04.010
Rodney Halburd, Risto Korhonen, and Kazuya Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4267–4298. MR 3206459, DOI 10.1090/S0002-9947-2014-05949-7
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
W. K. Hayman, On the characteristic of functions meromorphic in the plane and of their integrals, Proc. London Math. Soc. (3) 14a (1965), 93–128. MR 180679, DOI 10.1112/plms/s3-14A.1.93
Risto Korhonen, A new Clunie type theorem for difference polynomials, J. Difference Equ. Appl. 17 (2011), no. 3, 387–400. MR 2783355, DOI 10.1080/10236190902962244
Risto Korhonen, Nan Li, and Kazuya Tohge, Difference analogue of Cartan’s second main theorem for slowly moving periodic targets, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 523–549. MR 3525382, DOI 10.5186/aasfm.2016.4131
Ilpo Laine and Chung-Chun Yang, Clunie theorems for difference and $q$-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556–566. MR 2377111, DOI 10.1112/jlms/jdm073
A. Z. Mohon’ko, The Nevalinna characteristics of certain meromorphic functions, Teor. Funktsii Funktsional. Anal. i Prilozhen 14 (1971), 83–87.
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
Shun Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 2, 253–266. MR 632999
Georges Valiron, Sur la dérivée des fonctions algébroïdes, Bull. Soc. Math. France 59 (1931), 17–39 (French). MR 1504970, DOI 10.24033/bsmf.1170
Zhi-Tao Wen, Finite order solutions of difference equations, and difference Painlevé equations IV, Proc. Amer. Math. Soc. 144 (2016), no. 10, 4247–4260. MR 3531176, DOI 10.1090/proc/13210
Niro Yanagihara, Meromorphic solutions of some difference equations, Funkcial. Ekvac. 23 (1980), no. 3, 309–326. MR 621536
Jilong Zhang, Meromorphic solutions of difference Painlevé IV equations, Adv. Difference Equ. , posted on (2014), 2014:260, 12. MR 3347757, DOI 10.1186/1687-1847-2014-260
Jilong Zhang, Some results on difference Painlevé IV equations, J. Difference Equ. Appl. 22 (2016), no. 12, 1912–1929. MR 3592939, DOI 10.1080/10236198.2016.1255207
Jianhua Zheng, Value distribution of meromorphic functions, Tsinghua University Press, Beijing; Springer, Heidelberg, 2010. MR 2757285