Meromorphic functions sharing two small functions with its derivative

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM^∗, and share b(z)IM^∗ with g(z)=a₁(z)f(z)+a₂(z)f^′(z). And a₁(z), a₂(z) and b(z) (a₂(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.     

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.     

A bound on the multiplicity of horizontal sections for a connection on a Riemann surface

We address the question of bounding the multiplicity of the solutions of a linear differential system, setting the problem in invariant terms. A meromorphic connection is considered on a holomorphic vector bundle over a compact Riemann surface. We produce an upper bound on the order of vanishing of an arbitrary horizontal section, which depends only on global data, provided the connection has only regular singularities or the underlying monodromy is irreducible.     

分担超平面的截断型亚纯映射退化性定理

借鉴Yan和Chen在2011年以及Tan和Truong在2008年所采用的技巧和方法,给出了从C~m到P~n(C)的亚纯映射的截断型线性退化性定理.     

Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity

This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f(z) and its shift f(z+c), where c∈C, are studied. It is shown, for instance, that if f(z) is of finite order and shares two values CM and one value IM with its shift f(z+c), then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions, leading to a new way of characterizing elliptic functions. The research findings also include an analogue for shifts of a well-known conjecture by Brück concerning the value sharing of an entire function f with its derivative f^′.     

Meromorphic functions sharing values in an angular domain

In this paper we study the problems of meromorphic functions sharing values in an angular domain and obtain some uniqueness results which are the improvement of the well-known five-value theorem and four-value theorem given by Nevanlinna.     

On meromorphic extendibility

Let D be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint real-analytic simple closed curves. Let f be an integrable function on bD. In the paper we show how to compute the candidates for poles of a meromorphic extension of f through D and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility. Let A(D) be the algebra of all continuous functions on which are holomorphic on D. We prove that a continuous function f on bD extends meromorphically through D if and only if there is an N∈N∪{0} such that the change of argument of Pf+Q along bD is bounded below by −2πN for all P,Q∈A(D) such that Pf+Q≠0 on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity.     

The MacLane class and complex differential equations in the unit disk

The interaction between the coefficients and solutions for linear differential equations and for Riccati differential equations is investigated in the unit disk in terms of their asymptotic values as described by the MacLane class. This interaction is further explored within the context of a meromorphic version of the MacLane class.     

Logarithmic derivatives in annuli

In this paper, we define Nevanlinna functions in annuli with two independent variables and use them to prove a version of the generalized logarithmic derivative lemma in annuli. As a corollary, we can conclude the logarithmic derivative lemma in a disk and the complex plane.     

关于亚纯代数体函数奇异方向的存在性

本文证明了如下结果:对于有穷正级亚纯代数体函数,一定存在一条奇异方向L∶arg z=θ0(0≤θ0＜2π),使得对于任意δ∈(0,π/2),在角域Δ(θ0,δ)内,对任意复数a,对任意ε＞0,有∑i1|zi(a;Δ(θ0,δ))|σ=∞(σ等于ρ或ρ-ε)至多有2v个例外a值.