A note on partial sharing of values of meromorphic functions with their shifts

Charak, Kuldeep Singh; Korhonen, Risto; Kumar, G. Krishna

doi:10.1016/j.jmaa.2015.10.069

Cited by 24 publications

(10 citation statements)

References 10 publications

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“…Recently, the difference analogues to Nevanlinna's theory was established by Halburd and Korhonen [14,15], Chiang and Feng [8], independently, and improved by Halburd, Korhonen, and Tohge [16] from finite order of meromorphic functions to infinite order (hyper-order strictly less than 1). And it becomes a powerful theoretical tool to study the uniqueness problems of meromorphic functions taking into account their shifts (see, e.g., [3,12,13]) or difference operators (see, e.g., [18,22]), and so on. Due to the difference analogues to Nevanlinna's theory, Chen and his co-worker further discussed the above uniqueness problem in [4].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Meromorphic functions partially sharing 1CM+1IM concerning periodicities and shifts

Chun

Lü

2019

Filomat

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The purpose of this article is to deal with the uniqueness problems of meromorphic functions partially sharing values. It is showed that two entire functions f and with ρ 2 ( f ) < 1 and periodic restriction must be identically if E(0, f (z)) = E(0, (z)) except for a possible set G 1 and E(1, f (z)) = E(1, (z)) except for a possible set G 2 with N(r, G i ) = O(r λ ), (i = 1, 2), where λ(< 1) is a fixed constant. This result is a generalization of some previous works of Chen in [5] and Cai and Chen in [7].

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Meromorphic functions partially sharing 1CM+1IM concerning periodicities and shifts

Chun

Lü

2019

Filomat

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“…However, throughout the proof of the theorem Li-Yi [18] were tacit about the case when 0 is an e.v.p of f − a(z) or f − b(z) or both. From the previous results [2,11,22], over sharing of small functions for shift operator, it seems that the assumption about the periodicity of small functions is essential. In 2016, Charak-Korhonen-Kumar [2] exhibited the following example to claim that for shift operator of meromorphic function, "the case 1 CM + 2 IM (and hence 3 IM) does not hold in general"(see p. 1243, line 7 from top).…”

Section: Corollary 24 Under the Same Situation Of Theoremmentioning

confidence: 96%

Meromorphic functions of restricted hyper-order sharing small functions with their linear shift delay differential operator

Banerjee

Roy

2021

Rend. Circ. Mat. Palermo, II. Ser

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Taking into account of all the previous results, in this paper, we first resolve the "2 CM+1 IM" small functions sharing issue of a meromorphic function of restricted hyper order and its linear shift delay differential operator. We answer two open questions addressed by Qi-Yang (Analysis Math 46(4):843-865, 2020) in a convenient manner. Our second result, related to "2 IM" small function sharing problem of the same system, improves a result of (Analysis Math, 46(4):843-865, 2020). Moreover, concerning shift operator, we have pointed out a shortcoming in the sharing conditions in an example exhibited by Charak-Korhonen-Kumar (J Math Anal Appl 435(2): [1241][1242][1243][1244][1245][1246][1247][1248] 2016). In an attempt to resolve the issue, we have solved the "2 IM+1 CM" small functions sharing problem which in turn extend a result of Li-Yi (Bull Korean Math Soc 53(4): 2016).

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“…In recent years, the difference variant of the Nevanlinna theory has been established in [34][35][36][37]. Using these theories, some mathematicians began to consider the uniqueness of meromorphic functions sharing values with their shifts or difference operators and produced many fine works; for example, see Banerjee and Bhattacharyya [38], Ahamed [39,40], Ma et al [41], Jiang et al [42], Charak et al [43], Lin et al [44,45], and Li et al [46,47].…”

Section: Problem 1 (3im+1cm Open Problemmentioning

confidence: 99%

A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

Kang

Zhang

2021

Discrete Dynamics in Nature and Society

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In this paper, the 3IM+1CM theorem with a general difference polynomial L z , f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of L z , f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of L z , f is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of L z , f is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function f can be exactly expressed for the case f ≡ − L z , f . Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.

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A note on partial sharing of values of meromorphic functions with their shifts

Cited by 24 publications

References 10 publications

Meromorphic functions partially sharing 1CM+1IM concerning periodicities and shifts

Meromorphic functions partially sharing 1CM+1IM concerning periodicities and shifts

Meromorphic functions of restricted hyper-order sharing small functions with their linear shift delay differential operator

A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

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