Abstract
We prove a periodic theorem of meromorphic functions of hyper-order ρ2(f) < 1. As an application, we obtain the corresponding uniqueness theorem on periodic meromorphic functions. In addition, we show the accuracy of the results by giving some examples.
This work is supported by NSFs of China (No. 11371225), NSFs of Fujian Province (No. 2014J01004) and the Scientific Research Project of Fujian Provincial Education Department (No. JA15562, JA15394).The first author is supported by the Training Program of Excellent Talents in University(PETU) of Fujian Province in China as Junior Research Fellow in Tohoku University.
(Communicated by Stanisława Kanas)
References
[1] Brosch, G.: Eindeutigkeitssätze für Meromrophie Funktionen, Thesis, Technical University of Aachen, 1989.Search in Google Scholar
[2] Chiang, Y. M.—Feng, S. J.: On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J. 16 (2008), 105–129.10.1007/s11139-007-9101-1Search in Google Scholar
[3] Charak, K. S.—Korhonen R. J.—Kumar, G.: A note on partial sharing of values of meromorphic functions with their shifts, J. Math. Anal. Appl. 435 (2016), 1241–1248.10.1016/j.jmaa.2015.10.069Search in Google Scholar
[4] Chen, S. J.—Xu, A. Z.: Periodicity and unicity of meromorphic functions with three shared values, J. Math. Anal. Appl. 385 (2012), 485–490.10.1016/j.jmaa.2011.06.072Search in Google Scholar
[5] Chen, S. J.—Lin, W. C.: Periodicity and uniqueness of meromorphic functions concerning three sharing values, Houston J. Math. 43(3) (2017), 763–781.Search in Google Scholar
[6] Halburd, R. G.—Korhonen, R. J.—Tohge, K.: Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), 4267–4298.10.1090/S0002-9947-2014-05949-7Search in Google Scholar
[7] Halburd, R. G.—Korhonen, R. J.: Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), 463–478.Search in Google Scholar
[8] Heittokangas, J.—Korhonen, R.—Laine, I.—Rieppo J.—Zhang, J.: Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), 352–363.10.1016/j.jmaa.2009.01.053Search in Google Scholar
[9] Heittokangas, J.—Korhonen, R.—Laine, I.—Rieppo, J.: Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (2011), 81–92.10.1080/17476930903394770Search in Google Scholar
[10] Li, X. M.—Yi, H. X.: Meromorphic functions sharing four values with their difference operators or shifts, Bull. Korean Math. Soc. 53 (2016), 1213–1235.10.4134/BKMS.b150609Search in Google Scholar
[11] Ozawa, M.: On the existence of prime periodic entire functions, Kodai Math. J. 29 (1978), 308–321.10.2996/kmj/1138833654Search in Google Scholar
[12] Rubel, L. A.—Yang, C. C.: Values shared by an entire function and its derivative, In: Complex Analysis, Proc. Conf., Univ. Kentucky, Lexington, Ky. 1976, Lecture Notes in Math. 599, Springer, Berlin, 1977, pp. 101–103.10.1007/BFb0096830Search in Google Scholar
[13] Yi, H. X.—Yang, C. C.: Uniqueness Theory of Meromorphic Functions, Beijing Sci. Press, Beijing, 1995.Search in Google Scholar
[14] Yamanoi, K.: The second main theorem for small functions and related problems, Acta Math. 19(2) (2004), 225–294.10.1007/BF02392741Search in Google Scholar
[15] Zheng, J. H.: Unicity theorem for period meromorphic functions that share three values, Chi. Sci. Bull. 37(1) (1992), 12–15.10.2996/kmj/1138039600Search in Google Scholar
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