报告题目：Meromorphic solutions of the differentialequation $(f^{n})^{(k)}(g^{n})^{(k)}=1$ and its applications

报告人：李效敏教授 （中国海洋大学）

报告时间：2023年10月28日 16：00-17：00

报告地点：威廉希尔中文网站301

报告摘要：

In 1997, C.C.Yang and X.H.Hua [Ann. Acad. Sci. Fenn. Math.{\bf 22}(1997), no.2, 395-406] proved that if $f$ and $g$ are two nonconstant meromorphic functions such that $f^nf'g^n g'= 1,$ where $n$ is a positive integer satisfying $n\geq 6,$ then $f$ and $g$ are transcendental entire functions such that $g(z) = c_1e^{cz}$ and  $f(z) = c_2 e^{-cz},$ where $c,$ $c_1$ and $c_2$ are constants such that $(c_1 c_ 2 )^{ n+1}c^2 = -1.$ By Zalcman's Lemma, we prove that if $f$ and $g$ are two nonconstant meromorphic functions such that $(f^n)^{(k)}(g^n)^{(k)}=1,$ where $n$ and $k$ are positive integers satisfying $n>2k,$ then $f$ and $g$ are transcendental entire functions such that $f(z)=c_1e^{cz}$ and $g(z)=c_2e^{-cz},$ where $c_1,$ $c_2$ and $c$ are nonzero constants satisfying $(-1)^k(c_1c_2)^n(nc)^{2k}=1.$ Applying the result, we completely resolve a uniqueness question of meromorphic functions concerning certain nonlinear differential polynomials. As the applications of one of the main results in this paper, we also improve Theorem 1 from C.C.Yang and X.H.Hua [Ann.Acad.Sci.Fenn.Math.{\bf 22 }(1997), no.2, 395-406] and study a periodicity question of nonconstant meromorphic functions concerning certain nonlinear differential polynomials, where the periodicity question is related to a Yang's conjecture introduced in Q. Wang and P. C. Hu [Acta Math.Sci.{\bf 38}(2018), no.2, 209-214] and the differential-difference versions of the Yang's conjecture proposed in X. L. Liu and R. J. Korhonen[Bull. Australian Math. Soc. 101(2020), no.3, 453-465]. Our reasoning in this paper will make up the gap in the proof of Theorem 2 from S.S.Bhoosnurmath and R.S.Dyavanal [Comput. Math. Appl. {\bf 53}(2007), no. 8, 1191-1205].

报告人简介：李效敏，1967年1月2日出生，现任中国海洋大学数学系教授，硕士研究生指导教师，美国数学会数学评论员，德国Zentralblatt MATH 数据库评论员。威廉希尔中文网站87级插班生。1999年9月至2002年7月在山东大学数学系师从著名复分析专家仪洪勋教授攻读博士学位，于2002年7月在山东大学获得理学博士学位，并应聘到中国海洋大学数学系任教至今。近几年来本人主要从事复域上的微分方程理论、复差分方程理论、函数唯一性理论、解析数论中的L-函数值分布理论（包含黎曼ζ函数ζ(s)的零点分布）、代数体函数理论等复分析研究方向， 在《中国科学》、《Constructive Approximation》、《Ann. Fenn. Math.》(to appear)、《J.Math.Anal.Appl.》等国内外知名期刊发表学术论文70余篇，现已主持完成山东省自然科学基金面上项目两项，在研山东省自然科学基金面上项目一项，本人在山东大学攻读博士学位期间，曾多次参与仪洪勋教授主持的国家自然科学基金面上项目， 曾荣获山东省高等学校优秀科研成果奖二等奖、中国海洋大学天泰优秀人才奖。