Uniqueness of difference-differential polynomials of meromorphic functions sharing a small function IM

Ahamed, Molla Basir; Haldar, Goutam

doi:10.1007/s41478-021-00336-3

Cited by 7 publications

(5 citation statements)

References 18 publications

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“…Note that this possible only when gcd(n + m, n + 1) = 1 but which is not true if one consider some suitable value of n and m. For example if we choose n = 3 and m = 5, we note that gcd(n + m, n + 1) = gcd(8, 4) = 4 = 1. (iii) We observe that in [16, equation (49), page-950], the coefficient of T (r, g) is m n + m − 1 while actually it should be m n + m + 1 .…”

Section: Introduction Definitions and Main Resultsmentioning

confidence: 93%

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On uniqueness of two meromorphic functions sharing a small function

Ahamed

2019

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In this paper, we have investigated the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing a small function. Our results radically extended and improved the results of Bhoosnurmath-Pujari [9] and Harina -Anand [16] not only by sharing small function instead of fixed point but also reducing the lower bound of n. The authors Harina-Anand [16] made plenty of mistakes in their paper. We have corrected all of them in a more convenient way.At last some open questions have been posed for further study in this direction.

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Section: Introduction Definitions and Main Resultsmentioning

confidence: 93%

“…Thus we see that study on derivative or differential polynomial has a long history, several authors have been main engaged to find a possible relationship or certain forms of a function f when it shares small function with its derivative or differential polynomial (see [1] - [8]. )…”

Section: Introduction Definitions and Main Resultsmentioning

confidence: 99%

On uniqueness of two meromorphic functions sharing a small function

Ahamed

2019

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“…For finite ordered meromorphic functions, Halburd and Korhonen [18], and Chiang and Feng [14] developed independently parallel difference version of the famous Nevanlinna theory. As applications of this theory, we refer the reader to see the articles in case of set sharing problems (see, for example [3,4,7,6,10,37]), finding solutions to the Fermat-type difference equations (see e.g. [1,28,33]), Nevanlinna theory of the Askey-Wilson divided difference operators (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In 2014, Halburd et al [20] extended some results in this direction to meromorphic functions f whose hyper-order ρ 2 (f ) less than one. One may get much more information from [1,4,12,21,22,26,27] and the references therein, about the relationship between a meromorphic function f (z) and it shift f (z + c).…”

Section: Introductionmentioning

confidence: 99%

Class of meromorphic functions partially shared values with their differences or shifts

Ahamed¹

2021

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Two meromorphic functions f and g are said to share a value s ∈ C∪{∞} CM (IM ) provided that f (z) − s and g(z) − s have the same set of zeros counting multiplicities (ignoring multiplicities). We say that a meromorphic function f share s ∈ C partially CM with a meromorphic function g if E(s, f ) ⊆ E(s, g). It is easy to see that the condition "partially shared values CM " is more general than the condition "shared value CM ". With the idea of partially shared values, in this paper, we prove some uniqueness results between non-constant meromorphic functions and their shifts or generalized differences. We exhibit some examples to show that the result of Charak et al. [9] is not true for k = 2 or k = 3. We find some gaps in proof of the result of Lin et al.[25], and we not only correct them but also generalize their result in a more convenient way. A number of examples have been exhibited to validate certain claim of the main results of this paper and also to show that some of the conditions are sharp. In the end, we have posed some open questions for further investigation of the main result of the paper.

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“…Recently, with a more general setting, namely k-th derivative f (k) or differential monomial M [f ] or a more general setting namely differential polynomial P [f ], the present author, deduced that, when some power of a meromorphic function f and its differential monomial (or polynomial) sharing a set [1] or a small function [4], then even in this case, the function f also assumes a certain form. From the literature of meromorphic functions on sharing value problems, finding the class of the functions satisfying some differential or difference equations gained a valuable space.…”

Section: Introductionmentioning

confidence: 99%

A note on entire functions sharing a finite set with applications to difference equations

Ahamed

2020

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Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function f when sharing a set S = {α(z), β(z)} of two entire functions α and β with max{ρ(α), ρ(β)} < ρ(f ) with its difference L n c (f ) = n j=0 a j f (z + jc), then L n c (f ) ≡ f , and more importantly certain form of the function f has been found. The results in this paper improve those given by k. Liu, X. M. Li, J. Qi, Y. Wang and Y. Gu etc. Some examples have been exhibited to show the condition max{ρ(α), ρ(β)} < ρ(f ) is sharp in our main result. Examples have been also exhibited to show that if CM sharing is replaced by IM sharing, then conclusion of the main results ceases to hold.

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Uniqueness of difference-differential polynomials of meromorphic functions sharing a small function IM

Cited by 7 publications

References 18 publications

On uniqueness of two meromorphic functions sharing a small function

On uniqueness of two meromorphic functions sharing a small function

Class of meromorphic functions partially shared values with their differences or shifts

A note on entire functions sharing a finite set with applications to difference equations

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