Two meromorphic functions on annuli sharing some pairs of small functions or values

<abstract><p>In this paper, we prove that two admissible meromorphic functions on an annulus must be linked by a quasi-Möbius transformation if they share some pairs of small function with multiplicities truncated by $ 4 $. We also give the representation of Möbius transformation between two admissible meromorphic functions on an annulus if they share four pairs of values with multiplicities truncated by $ 4 $. In our results, the zeros with multiplicities more than a certain number are not needed to be counted if their multiplicities are bigger than a certain number.</p></abstract>

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TWO MEROMORPHIC FUNCTIONS SHARING SOME PAIRS OF SMALL FUNCTIONS REGARDLESS OF MULTIPLICITIES

International Journal of Mathematics ◽

10.1142/s0129167x14500141 ◽

2014 ◽

Vol 25 (02) ◽

pp. 1450014 ◽

Cited By ~ 1

Author(s):

SI DUC QUANG ◽

LE NGOC QUYNH

Keyword(s):

Meromorphic Functions ◽

Möbius Transformation ◽

Small Functions ◽

Mobius Transformation

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.

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Meromorphic Functions Sharing Pairs of Small Functions

Mathematica Slovaca ◽

10.1515/ms-2015-0009 ◽

2015 ◽

Vol 65 (1) ◽

Author(s):

Jilong Zhang ◽

Lianzhong Yang

Keyword(s):

Meromorphic Functions ◽

Möbius Transformation ◽

Small Functions ◽

Mobius Transformation

AbstractIn this paper, we investigate the relations between two meromorphic functions that share some pairs of small functions. In particular, we show that two meromorphic functions that share six pairs of small functions IM∗ must be linked by quasi-Möbius transformation. We also give some properties of two meromorphic functions sharing five pairs of values or small functions.

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Finiteness problem of meromorphic functions sharing four small functions regardless of multiplicities

International Journal of Mathematics ◽

10.1142/s0129167x1450102x ◽

2014 ◽

Vol 25 (11) ◽

pp. 1450102

Author(s):

Si Duc Quang

Keyword(s):

Meromorphic Functions ◽

Small Function ◽

Small Functions

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.

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UNICITY OF MEROMORPHIC FUNCTIONS SHARING SOME SMALL FUNCTIONS REGARDLESS OF MULTIPLICITIES

International Journal of Mathematics ◽

10.1142/s0129167x12500887 ◽

2012 ◽

Vol 23 (09) ◽

pp. 1250088 ◽

Cited By ~ 2

Author(s):

SI DUC QUANG

Keyword(s):

Meromorphic Functions ◽

Small Function ◽

Finiteness Theorem ◽

Unicity Theorem ◽

Small Functions

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.

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