Narrow Results

The purpose of this article is twofold. The first is to prove the unicity theorem with truncated multiplicities of meromorphic functions sharing five small functions. This gives a remarkable improvement of the results of Yuhua–Jianyong, Yao and Yi. The second is to generalize the unicity theorem of Fujimoto to meromorphic functions sharing four small functions with truncated multiplicities.

The purpose of this article is twofold. The first is to prove a unicity theorem for meromorphic functions sharing five small functions regardless of multiplicities. This is an improvement of the results of Yahua–Jianyong, Yao and Yi, Thai and Tan. The second is to prove a finiteness theorem for meromorphic functions that share three small functions ignoring multiplicity and another small function with multiplicities truncated by 2.

In this paper, we prove that two meromorphic functions f and g must be linked by a quasi-Möbius transformation if they share a pair of small functions ignoring multiplicities and share other four pairs of small functions with multiplicities truncated by 2. We also show that two meromorphic functions which share q (q ≥ 6) pairs of small functions ignoring multiplicities are linked by a quasi-Möbius transformation. Moreover, in our results, the zeros with multiplicities more than a certain number are not needed to be counted in the condition sharing pairs of small functions of meromorphic functions. These results are generalization and improvements of some recent results.

This paper has twofolds. The first is to prove that there are at most two meromorphic functions sharing a small function with multiplicities truncated by 2 and other three small functions regardless of multiplicities, where all zeros with multiplicities more than a certain number are not counted. This result is an improvement of the four-value theorems of Nevanlinna, Gundersen, Fujimito, Thai–Tan and others. The second purpose of this paper is to prove that there are at most three meromorphic functions sharing four small functions ignoring multiplicity, where all zeros with multiplicities more than a certain number are omitted.

In this paper, we investigate the uniqueness problems of meromorphic functions concerning differential polynomials sharing a small function. In particular, the resuts of the paper improve the results due to Waghamore and Anand [11].

In this paper, we mainly discuss the uniqueness problem of k order derivatives of meromorphic functions that share four small functions, and consider Nevanlinna's Four Values Theorem when the derivatives of meromorphic functions share some small functions, which generalize and improve some related results of the author.

In this paper, a new singular direction of meromorphic function namely T-direction is investigated. Let a meromorphic function takes the value of its small function, we prove the existence of the corresponding T-direction.

In this paper, we proved a result that if two meromorphic functions f(z) and g(z) share four small functions a_j (z)(j = 1,⋯, 4) in the sense of E_k)(a_j, f) = E_k)(a_j,g), (j = 1, ⋯, 4) (k ≥ 11), then f is a quasi-Möbius transformation of g; i.e., there exist four small functions α_i(i = 1, ⋯, 4) of f and g such that f = (α₁g + α₂)/(α₃g + α₄), where α₁α₄− α₂α₃ ≢ 0.