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
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations
Articles in the same Issue
- Prof. RNDr. pavel brunovský, DrSc. passed away ∗dec. 5, 1934 – † dec. 14, 2018
- Ideals and congruences in pseudo-BCH algebras
- Regular double p-algebras
- Distributive nearlattices with a necessity modal operator
- States in generalized probabilistic models: An approach based in algebraic geometry
- Free power-associative n-ary groupoids
- On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions
- On the space of generalized theta-series for certain quadratic forms in any number of variables
- Uniqueness and periodicity for meromorphic functions with partial sharing values
- Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups
- An inverse boundary problem for fourth-order Schrödinger equations with partial data
- Entropy as an integral operator
- Global behavior of two third order rational difference equations with quadratic terms
- Measures on effect algebras
- Some topological and combinatorial properties preserved by inverse limits
- The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces
- Compositions of porouscontinuous functions
- A note on prime divisors of polynomials P(Tk); k ≥ 1
- On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications
- Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations