Abstract
We present some results on the periodicity of an entire function f(z) with its differential-difference polynomials. For instance, we obtain that if fn + L(f) is a periodic function with period c, then f(z) must be a periodic function with period c, where f(z) is a transcendental entire function with hyper-order less than one and \(N(r,{1 \over f}) = S(r,f),\,L(f)\) is a differential-difference polynomial in f with constant coefficients and degree less than n.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
X. J. Dong and L. W. Liao, Meromorphic solutions of certain type of differential-difference equations, J. Math. Anal. Appl., 470 (2019), 327–339.
R. G. Halburd, R. J. Korhonen and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc., 366 (2014), 4267–4298.
W. K. Hayman, Meromorphic Functions, Clarendon Press (Oxford, 1964).
W. K. Hayman, Waring’s problem für analytische Funktionen, Bayer. Akad. Wiss. Math. Natur. Kl. Sitzungsber., 1984 (1985), 1–13.
J. Heittokangas, R. J. Korhonen and I. Laine, On meromorphic solutions of certain nonlinear differential equations, Bull. Aust. Math. Soc., 66 (2002), 331–343.
J. Heittokangas, Z. Latreuch, J. Wang and M. A. Zemirni, On meromorphic solutions of non-linear differential equations of Tumura—Clunie type, Math. Nachr., 294 (2021), 748–773.
Z. Latreuch and M. A. Zemirni, On a theorem of A. and C. Rényi and a conjecture of C. C. Yang concerning periodicity of entire function, Anal. Math., 48 (2022), xx–yy.
P. Li, W. R. Lü and C. C. Yang, Entire solutions of certain types of nonlinear differential equations, Houston J. Math., 45 (2019), 431–437.
X. L. Liu and R. Korhonen, The periodicity of transcendental entire functions, Bull. Aust. Math. Soc., 101 (2020), 453–465.
X. L. Liu, R. Korhonen and K. Liu, Variations on a Conjecture of C. C. Yang concerning periodicity, Comput. Methods Funct. Theory, https://doi.org/10.1007/s40315-020-00359-0.
K. Liu, Y. M. Wei and P. Y. Yu, Generalized Yang’s Conjecture on the periodicity of entire functions, Bull. Korean Math. Soc., 57 (2020), 1259–1267.
K. Liu and P. Y. Yu, A note on the periodicity of entire functions, Bull. Aust. Math. Soc., 100 (2019), 290–296.
W. R. Lü, L. L. Wu, D. D. Wang and C. C. Yang, The existence of solutions to certain type of non-linear difference-differential equations, Open Math., 16 (2018), 806–815.
W. R. Lü and X. X. Zhang, On the periodicity of entire functions, Results Math., 75 (2020), Paper No. 176, 13 pp.
A. Rényi and C. Rényi, Some remarks on periodic entire functions, J. Analyse Math., 14 (1965), 303–310.
Q. Wang and P. C. Hu, On zeros and periodicity of entire functions, Acta. Math. Sci. Ser. A (Chin. Ed.), 38 (2018), 209–214.
C. C. Yang, On entire solutions of a certain type of nonlinear differential equations, Bull. Aust. Math. Soc., 64 (2001), 377–380.
C. C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A, 86 (2010), 10–14.
C. C. Yang and P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math., 82 (2004), 442–448.
C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers (2003).
Acknowledgements
Authors would like to thank Prof Kai Liu and the reviewer for useful comments and discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the NSFC (No. 12061042) and by Jiangxi Provincial Natural Science Foundation (No. 20202BAB201003).
Rights and permissions
About this article
Cite this article
Liu, X.L., Wei, Y.M. The periodicity of differential-difference polynomials of entire functions. Anal Math 48, 127–138 (2022). https://doi.org/10.1007/s10476-022-0119-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10476-022-0119-9