# Five-value rich lines‎, ‎Borel directions and uniqueness of meromorphic functions

Document Type: Research Paper

Author

School of Mathematical Sciences‎, ‎Guizhou Normal University‎, ‎550001‎, ‎Guiyang‎, ‎P.R‎. ‎China ‎School of Computer Sciences and School of Sciences‎, ‎Beijing University of Posts and Telecommunications‎, ‎Beijing‎, ‎100876‎, ‎P.R‎. ‎China‎.

Abstract

For a meromorphic function $f$ in the complex plane, we shall introduce the definition of five-value rich line of $f$, and study the uniqueness of meromorphic functions of finite order in an angular domain by involving the five-value rich line and Borel directions. Finally, the relationship between a five-value rich line and a Borel direction is discussed, that is, every Borel direction of $f$ is its five-value rich line, and the inverse statement holds when $f$ is of infinite order.

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### References

T.B. Cao and H.X. Yi, On the uniqueness of meromorphic functions that share four values in one angular domain, J. Math. Anal. Appl. 358 (2009), no. 1, 81--97.

C.T. Chuang, Sur les fonctions-types, Sci. Sinica 10 (1961) 171--181.

C.T. Chuang, Singular Directions of Meromorphic Functions (Chinese), Science Press, Beijing, 1982.

C.T. Chuang, On Borel directions of meromorphic functions of infinite order (⨿), Bull. Hong Kong Math. Soc. 2 (1999), no. 2, 305--323.

D. Drasin and A. Weitsman, On the Julia directions and Borel directions of entire functions, Proc. London Math. Soc. (3) 32 (1976), no. 3, 199--212.

A. Eremenko, I.V. Ostrovskii and M. Sodin, Anatolii Asirovich Goldberg, Complex Var. Theory Appl. 37 (1998), no. 1, 1--51.

A.A. Goldberg and I.V. Ostrovskii, Value Distribution of Meromorphic Functions, Transl. Math. Monogr. 236, Amer. Math. Soc. Providence, RI, 2008.

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 (1935) 233--308.

W.C. Lin, S. Mori and K. Tohge, Uniqueness theorems in an angular domain, Tohoku Math. J. (2) 58 (2006), no. 4, 509--527.

W.C. Lin, S. Mori and H.X. Yi, Uniqueness theorems of entire functions with shared-set in an angular domain, Acta Math. Sinica 24 (2008), no. 11, 1925--1934.

J.R. Long and C.H. Qiu, Borel directions and uniqueness of meromorphic functions sharing ve values, J. Comput. Anal. Appl. 21 (2016), no. 6, 1046--1059.

J.R. Long and P.C. Wu, Borel directions and uniqueness of meromorphic functions (Chinese), Chinese Ann. Math. Ser. A 33 (2012), no. 3, 261--266.

R.H. Nevanlinna, Le Theoreme de Picard-Borel et la Theorie de Fonctions Meromorphes, Gauthier-Villars et Cie, Paris, 1929.

M. Tsuji, On Borel directions of meromorphic functions of finite order, I, Tohoku Math. J. (2) 2 (1950) 97--112.

S.J. Wu, The distribution of Borel directions of entire functions (Chinese), Chinese Ann.Math. Ser. A 14 (1993), no. 4, 400--406.

S.J. Wu, Further results on Borel removable sets of entire functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 19 (1994), no. 1, 67--81.

J.F. Xu and H.X. Yi, On uniqueness of meromorphic functions with shared four values in some angular domains, Bull. Malays. Math. Sci. Soc. (2) 31 (2008), no. 1, 57--65.

L. Yang, Borel directions of meromorphic functions in an angular domain, Sci. Sinica 1979, Special Issue I on Math., 149--164.

L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.

L. Yang and C.C. Yang, Angular distribution of values of ff, Sci. China Ser. A 37 (1994), no. 3, 284--294.

C.C. Yang and H.X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications 557, Kluwer Acad. Publ. Dordrecht, 2003.

L. Yang and G.H. Zhuang, The distribution of Borel directions of entire functions, Sci. China Ser. A 3 (1976) 157--168.

Q.C. Zhang, Meromorphic functions sharing values in an angular domain, J. Math. Anal. Appl. 349 (2009), no. 1, 100--112.

J.H. Zheng, On uniqueness of meromorphic functions with shared values in one angular domain, Complex Var. Theory Appl. 48 (2003), no. 9, 777--785.

J.H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad. Math. Bull. 47 (2004), no. 1, 152--160.

### History

• Receive Date: 31 October 2015
• Revise Date: 17 September 2016
• Accept Date: 17 September 2016