{"title":"Zeros of Differential Polynomials of Meromorphic Functions","authors":"Ta Thi Hoai An, Nguyen Viet Phuong","doi":"10.1007/s40306-021-00442-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>f</i> be a transcendental meromorphic function on <span>\\(\\mathbb {C},\\)</span> <i>k</i> be a positive integer, and <span>\\(Q_{0},Q_{1},\\dots ,Q_{k}\\)</span> be polynomials in <span>\\(\\mathbb {C}[z]\\)</span>. In this paper, we will prove that the frequency of distinct poles of <i>f</i> is governed by the frequency of zeros of the differential polynomial form <span>\\(Q_{0}(f)Q_{1}(f^{\\prime }){\\dots } Q_{k}(f^{(k)})\\)</span> in <i>f</i>. We will also prove that the Nevanlinna defect of the differential polynomial form <span>\\(Q_{0}(f)Q_{1}(f^{\\prime }){\\dots } Q_{k}(f^{(k)})\\)</span> in <i>f</i> satisfies \n</p><div><div><span>$$ \\sum\\limits_{a\\in\\mathbb{C}}\\delta\\left( a,Q_{0}(f)Q_{1}(f^{\\prime}){\\dots} Q_{k}\\left( f^{(k)}\\right)\\right)\\leq 1$$</span></div></div><p>\nwith suitable conditions on <i>k</i> and the degree of the polynomials. Thus, our work is a generalization of Mues’s conjecture and Goldberg’s conjecture for the more general differential polynomials.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40306-021-00442-1","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00442-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let f be a transcendental meromorphic function on \(\mathbb {C},\)k be a positive integer, and \(Q_{0},Q_{1},\dots ,Q_{k}\) be polynomials in \(\mathbb {C}[z]\). In this paper, we will prove that the frequency of distinct poles of f is governed by the frequency of zeros of the differential polynomial form \(Q_{0}(f)Q_{1}(f^{\prime }){\dots } Q_{k}(f^{(k)})\) in f. We will also prove that the Nevanlinna defect of the differential polynomial form \(Q_{0}(f)Q_{1}(f^{\prime }){\dots } Q_{k}(f^{(k)})\) in f satisfies
with suitable conditions on k and the degree of the polynomials. Thus, our work is a generalization of Mues’s conjecture and Goldberg’s conjecture for the more general differential polynomials.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.