Abstract
Let f be a nonconstant meromorphic function of lower order µ (f) > 1/2 in ℂ, and let aj (j = 1, 2, 3) be three distinct finite complex numbers. We show that there exists an angular domain D = {z: α ≤ arg z ≤ β}, where 0 < β − α ≤ 2π, such that if f share aj (j = 1, 2, 3) CM with its k-th linear differential polynomial L[f] in D, then f = L[f]. This generalizes the corresponding results from Frank and Schwick, Zheng and Li-Liu-Yi.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.T. Chuang, “Sur les fonctions-types”, Sci. Sinica, 10, 171–181, 1961.
A. Eremenko, I.V. Ostrovskii, and M. Sodin, “Anatolii Asirovich Gol’dberg”, Complex Var. Theory Appl., 37(1–4), 1–51, 1998.
G. Frank, W. Schwick, “Meromorphe Funktionen, die mit einer Ableitung drei Werte teilen”, Results. Math., 22, 679–684, 1992.
G. Frank, G. Weissenborn, “Meromorphe Funktionen, die mit einer ihrer Ableitungen Werte teilen”, Complex Var., 7, 33–43, 1986.
A.A. Goldberg, I.V. Ostrovskii, Distribution of values of meromorphic functions (Nauka, Moscow, 1970).
G.G. Gundersen, “Meromorphic functions that share finite values with their derivatives”, J. Math. Anal. Appl., 75, 441–446, 1980.
W.K. Hayman, Meromorphic Functions (Clarendon Press, Oxford, 1964).
K.L. Hiong, “Sur les fonctions entieres et les fonctions meromorphes d’ordre infini”, J. Math. Pures Appl., 14, 233–308, 1935.
X.M. Li, C. Liu, H.X. Yi, “Meromorphic functions sharing three values with their linear differential polynomials in some angular domains”, Results. Math., 68, 441–453, 2015.
E. Mues and N. Steinmetz, “Meromorphe funktionen, die mit ihrer ableitung werte teilen”, Manuscripta Math., 29, 195–206, 1979.
R. Nevanlinna, “Über die eigenschaften meromorpher funktionen in einem winkelraum”, Acta Soc. Sci. Fenn., 50, 1–45, 1925.
G. Valiron, “Recherches sur de théorème de M. Borel dans la théorie des fontions méromorphes”, Acta Math., 52, 67–92, 1928.
G. Valiron, “Sur les directions de Borel des fonctions meéromorphes d’ordre infini”, C. R. Acad. Sci. Paris Sér. I Math., 206, 575–577, 1938.
L. Yang, Value distribution theory (Springer, Berlin, 1993).
H.X. Yi and C.C. Yang, Uniqueness theory of meromorphic functions, Pure and Applied Math. Monographs No. 32 (Science Press, Beijing, 1995).
J.H. Zheng, “On uniqueness of meromorphic functions with shared values in some angular domains”, Canad. J. Math., 47, 152–160, 2004.
Acknowledgments
The author would like to thank the referees for their thorough comments and helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2019, No. 3, pp. 35–44.
The research was supported by the National Natural Science Foundation of China (Grant No. 11301076), and by the Natural Science Foundation of Fujian Province, China (Grant No. 2018J01658).
About this article
Cite this article
Chen, JF. Meromorphic Functions Sharing Three Values with Their Linear Differential Polynomials in an Angular Domain. J. Contemp. Mathemat. Anal. 54, 157–162 (2019). https://doi.org/10.3103/S106836231903004X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S106836231903004X