Meromorphic functions sharing values in an angular domain

Abstract

In this paper we study the problems of meromorphic functions sharing values in an angular domain and obtain some uniqueness results which are the improvement of the well-known five-value theorem and four-value theorem given by Nevanlinna.

Nevanlinna’s Five Values Theorem on Annuli

“…Zhang, T.B. Cao and W.C. Lin (including [12][13][14][15][16]) considered many uniqueness problem on meromorphic functions on the angular domain.…”

Meromorphic functions in the unit disc that share values in an angular domain

“…Zheng firstly took into account the uniqueness of meromorphic functions sharing values in an angular domain and extended fivevalue theorem and four-value theorem in the complex plane to an angular domain (see [5,6]). Since then, the uniqueness of meromorphic functions in an angular domain attracted many investigations (see [7][8][9], etc.). In fact, he proved the following theorems on the basis of the relation between the Pólya peaks and deficiencies of meromorphic functions.…”

“…Zhang [9] proved the following theorems under giving some restrictions on the angular characteristic functions of meromorphic functions.…”

“…Theorem C. (See [9].) Let f (z) and g(z) be two meromorphic functions of finite order in C, a j ∈ C ∞ ( j = 1, 2, .…”

“…Theorem D. (See [9].) Let f (z) and g(z) be two meromorphic functions of finite order in C, a j ∈ C ∞ ( j = 1, 2, 3, 4) be four distinct values, and let δ = {z | |arg z − θ 0 | δ} (0 < δ < π) be an angular domain satisfying (1.1).…”