Second Main Theorem and Uniqueness Problem of Meromorphic Functions with Finite Growth Index Sharing Five Small Functions on a Complex Disc

Abstract

This paper has a twofold purpose. The first is to establish a second main theorem for meromorphic functions on the complex disc Δ(R₀) ⊂ ℂ with finite growth index and small functions, where the counting functions are truncated to level 1 and the small term is more detailedly estimated. The second is to prove a generalization and improvement of the five values theorem of Nevanlinna for the case of five small functions on the complex disc Δ(R₀).

Keywords

Nevanlinna theory / Second main theorem / Meromorphic function / Small function / 30D35 / 32H30 / 32A22

References

[1] Nevanlinna R. Einige Eideutigkeitssäte in der théorie der meromorphen funktionen. Acta. Math., 1926, 48: 367-391.

[2] Quang S D. Unicity of meromorphic functions sharing some small function. Internat. J. Math., 2012, 23(9): 1250088.

[3] Ru M, Sibony N. The second main theorem in the hyperbolic case. Math. Annalen, 2020, 377: 759-795.

[4] Yamanoi K. The second main theorem for small functions and related problems. Acta Math., 2004, 192: 225-294.

[5] Yao W. Two meromorphic functions sharing five small functions in the sense ${\overline E}_{k)(\beta, \, f)} = {\overline E}_{k)}{(\beta, \, g)}$. Nagoya Math. J., 2002, 167: 35-54.

[6] Yi H-X. On one problem of uniqueness of meromorphic functions concerning small functions. Proc. Amer. Math. Soc., 2002, 130: 1689-1697.

[7] Yi H-X, Yang C C. Uniqueness Theory of Meromorphic Functions, 1995. Beijing, Science Press. 32

[8] Yuhua L, Jianyong Q. On the uniqueness of meromorphic functions concerning small functions. Sci. China Ser. A, 1999, 29: 891-900

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg