This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970.
We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f.We then characterize meromorphic functions admitting a harmonic star function.
Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of $f$ instead of its Wronskian determinant. As its applications, we obt…
In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.
In this article, we prove that there are at most two meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 2)$ sharing $2n+2$ hyperplanes in general position regardless of multiplicity, where all zeros with multiplicities more than certain values do not need to be counted. We also show that if three meromorphic mappings $f^1,f^2,f^3$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 5)$ share $2n+1$ hyperplanes in general position with truncated multiplicity then the map $f^1\times f^2\times f^3$ is linearly degenerate.
We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.
This is a survey of results on the following problem. Consider a simply connected Riemann surface spread over the Riemann sphere. How are the properties of the uniformizing function of this surface related to the geometric properties of the surface? Based on the lecture in U. Michigan in May 2006.
In this paper, we mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables, and then investigate meromorphic solutions of partial difference equations from the viewpoint of Nevanlinna theory.
An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $πi$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We improves the author's result \cite{h} from entire function to meromorphic function, the first derivative to its differential-difference polynomial, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $ρ_{2}(f)<1$, l…
Meromorphic germs in several variables with linear poles naturally arise in mathematics in various disguises. We investigate their rich structures under the prism of locality, including locality subalgebras, locality transformation groups and locality characters. The key technical tool is the dependence subspace for a meromorphic germ with which we define a locality orthogonal relation between two meromorphic germs. We describe the structure of locality subalgebras generated by classes of meromorphic germs with certain types of poles. We also define and determine their group of locality transf…
This is an expanded version of one of the Lectures in memory of Lars Ahlfors in Haifa in 1996. Some mistakes are corrected and references added. It contains a survey of his work on meromorphic functions and related topics written in 1929-1941.
The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By using the theorem, we reprove some results about zeros of derivative of Xi function,Gamma function and digamma function in a new way.
We study one variable meromorphic functions mapping a planar real algebraic set $A$ to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain $A$, these meromorphic functions must be rational. In particular, when $A$ is the standard unit circle, we obtain an one dimensional analog of Poincaré(1907), Tanaka(1962) and Alexander(1974)'s rationality results for $2m-1$ dimensional sphere in $\mathbb{C}^m$ when $m\ge 2$.
In this note, it is shown that the differential polynomial of the form $Q(f)^{(k)}-p$ has infinitely many zeros, and particularly $Q(f)^{(k)}$ has infinitely many fixed points for any positive integer $k$, where $f$ is a transcendental meromorphic function, $p$ is a nonzero polynomial and $Q$ is a polynomial with coefficients in the field of small functions of $f$. The results are traced back to Problem 1.19 and Problem 1.20 in the book of research problems by Hayman and Lingham. As a consequence, we give an affirmative answer to an extended problem on the zero distribution of $(f^n)'-p$, prop…
A survey of general results on the singularities of inverses to meromorphic functions is given, with applications to holomorphic dynamics. This is a lecture delivered at the workshop "The role of complex analysis in complex dynamics" in Edinburgh on May 22 2013, with corrected mistakes and updated references.
Meromorphic solutions of first-order differential equations with rational exponential coefficients
Institute of Applied Mathematics, Shenzhen Polytechnic University（深圳职业技术大学应用数学研究所）；School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；School of Basic and General Education, Guangzhou University of Software（广州软件学院基础与通识教育学院）
Backtracking New Q-Newton's method for finding roots of meromorphic functions in 1 complex variable: Global convergence, and local stable/unstable curves
We use contact geometry to describe the monoid of projectively equivariant meromorphic differential operators on a complex curve, quantization of which generalizes known constructions of classical equivariants to non-commutative function algebras in several variables.
This is a survey talk on the ICMS conference "Resonances of complex dynamics", 9-13 July 2018. Three examples which show how dynamical considerations lead to new results in complex function theory are discussed. These examples are: a) a refinement of of the Wiman-Valiron method, b) the proof of Hayman's conjecture on the value distribution of derivatives of meromorphic functions, and c) a new simplified proof of the result of Lin and Wang on the number of critical points of the Green function on a torus.
The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function
Laboratoire de Mathématiques UMR 8100, CNRS, Université de Versailles-St Quentin（凡尔赛圣康坦大学数学实验室）；Dipartimento di Matematica Guido Castelnuovo, Università Sapienza（罗马第一大学圭多·卡斯泰尔努沃数学系）
本文证明了 Euler gamma 函数的 Ax-Lindemann-Weierstrass 型微分超越结果：对于模 Z 两两不同且代数于亚纯 1-周期函数域 k 的亚纯函数 ζ_i，函数 Γ(ν-ζ_i(ν)) 在 k(ν) 上微分独立。
Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta f…
