Uniqueness of Meromorphic Functions With Nonlinear Differential Polynomials Sharing a Small Function IM

Q3 Mathematics
H. R. Jayarama, S. Bhoosnurmath, C. N. Chaithra, S. Naveenkumar
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引用次数: 0

Abstract

In the paper, we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain some new results regarding the structure and position of uniqueness. These new results have immense applications like classifying different expressions to be or not to be unique. The principal objective of the paper is to study the uniqueness of meromorphic functions when sharing a small function $a(z)$ IM with restricted finite order and its nonlinear differential polynomials. The lemma on the logarithmic derivative by Halburb and Korhonen (Journal of Mathematical Analysis and Applications, \textbf{314} (2006), 477--87) is the starting point of this kind of research. In this direction, the current focus in this field involves exploring unique results for the differential-difference polynomials of meromorphic functions, covering both derivatives and differences. Liu et al. (Applied Mathematics A Journal of Chinese Universities, \textbf{27} (2012), 94--104) have notably contributed to this research. Their research establishes that when $n \leq k + 2$ for a finite-order transcendental entire function $f$ the differential-difference polynomial$[f^{n}f(z+c)]^{(k)} - \alpha(z)$ has infinitely many zeros. Here, $\alpha(z)$ is characterized by its smallness relatively to $f$. Additionally, for two distinct meromorphic functions $f$ and $g$, both of finite order, if the differential-difference polynomials $[f^{n}f(z+c)]^{(k)}$\ and\ $[g^{n}g(z+c)]^{(k)}$ share the value $1$ in the same set, then $f(z)=c_1e^{dz},$ $g(z)=c_2e^{-dz}.$ We prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J., \textbf{71} (2019), 1032--1042), and Zhang and Xu (Comput. Math. Appl., \textbf{61} (2011), 722-730) and citing a proper example we have shown that the result is true only for a particular case. Finally, we present the compact version of the same result as an improvement.
具有共享小函数 IM 的非线性微分多项式的同调函数的唯一性
在本文中,我们从不同的多项式角度讨论了唯一性及其元素在扩展复平面上的分布。我们获得了一些关于唯一性的结构和位置的新结果。这些新结果具有巨大的应用价值,比如对不同表达式的唯一性或非唯一性进行分类。本文的主要目的是研究当共享具有受限有限阶的小函数 $a(z)$ IM 及其非线性微分多项式时,分形函数的唯一性。Halburb 和 Korhonen 关于对数导数的 Lemma (Journal of Mathematical Analysis and Applications, \textbf{314} (2006), 477--87) 是此类研究的起点。在这个方向上,这一领域目前的重点是探索微分-差分多项式的独特结果,包括导数和差分。Liu et al.(2012), 94--104)对这一研究做出了突出贡献。他们的研究证明,当 $n \leq k + 2$ 为有限阶超越全函数 $f$ 时,微分-差分多项式$[f^{n}f(z+c)]^{(k)} - \alpha(z)$ 有无穷多个零点。这里,$\alpha(z)$ 的特征是相对于 $f$ 而言它很小。此外,对于两个不同的有限阶函数 $f$ 和 $g$,如果微分-差分多项式 $[f^{n}f(z+c)]^{(k)}$ 和 $[g^{n}g(z+c)]^{(k)}$在同一个集合中共享值 $1$,那么 $f(z)=c_1e^{dz},$g(z)=c_2e^{-dz}。$ 我们证明了两个结果,它们极大地推广了 Dyavanal 和 Mathai 的结果 (Ukrainian Math. J., \textbf.J., \textbf{71} (2019), 1032--1042), 以及 Zhang 和 Xu (Comput.Math.Appl., \textbf{61} (2011), 722-730),并列举了一个适当的例子,证明了这一结果只在特定情况下才是正确的。最后,我们提出了同一结果的精简版本,作为一种改进。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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