Meromorphic functions

The author surveys meromorphic functions, with primary emphasis on the distribution of values; the superposition and iteration of meromorphic functions are reviewed, along with meromorphic solutions of functional, differential, and difference equations. The survey covers the period from 1953 through 1971.Translated from Itogi Nauki i Tekhniki (Matematicheskii Analiz), Vol. 10, pp. 5–97, 1973.     

Meromorphic functions sharing small functions

Let f and g be meromorphic functions sharing four small functions a₁, a₂, a₃, a₄ a_1, a_2, a_3, a_4 ignoring multiplicities. If there is a small function a₅a_5 distinct from a_j, j=1, 2, 3, 4, a_j, j=1, 2, 3, 4, such that [`(N)](r,f=a<sub>5</sub>=g) 1 S(r,f) overline {N}(r,f=a_5=g)ne S(r,f) , then f=g f=g , where [`(N)](r,f=a₅=g) overline{N}(r,f=a_5=g) is the counting function of those common zeros of f(z)-a₅(z) f(z)-a_5(z) and g(z)-a₅(z) g(z)-a_5(z) counted only once ignoring multiplicities.     

Meromorphic functions with bounded boundary rotation

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

具有重值分担的亚纯函数

在考虑重值分担的情况下,得到一个具有三个分担值的亚纯函数的唯一性定理及若干推论,推广并改进了仪洪勋、Lah iri等人的结果.     

与其导数分担三个小函数的亚纯函数

主要得到亚纯函数与其导数分担三个小函数时的一个唯一性定理.     

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.     

Meromorphic functions sharing four small functions

It is well known that if two nonconstant meromorphic functions f and g on the complex plane ? have the same inverse images counted with multiplicities for four distinct values, then g is a Möbius transformation of f. In this paper, we will show that the above result remains valid if f and g share four distinct small functions counted with multiplicities truncated by 2. This is the best possible truncation level.     

Meromorphic functions and factoriality

Let be a compact subset of a connected Stein manifold . We study algebraic properties of the ring of meromorphic functions on without poles in .

     

具三个公共小函数的亚纯函数

讨论具三个公共小函数的亚纯函数的唯一性问题,其结果推广并改进了H.Ueda的一个定理。     

Meromorphic functions with weighted sharing of three values

In this article we study the problem of uniqueness of meromorphic functions with weighted sharing of three values. The results in this article improve some theorems of M. Ozawa, H. Ueda, G. Brosch, H. Yi, I. Lahiri and other authors. Examples are provided to show that the conditions of our results are sharp.     

Meromorphic functions with deficiencies generating unique range sets

With the help of weighted sharing of sets we deal with the problem of unique range set for meromorphic functions with deficient values and obtain a result which improves, generalizes and extends some previous results. We provide two examples to show that the condition in one of our results is the best possible.     

Meromorphic functions sharing two sets

In the paper we discuss the uniqueness problem for meromorphic functions that share two sets and prove five theorems which improve and supplement some results earlier given by Yi and Yang [13], Lahiri and Banerjee [5].     

On Gibson functions with connected graphs

A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f(\bar G) \subseteq \overline {f(G)} $ for any open connected set G ? X. We call a function f: X → Y segmentary connected if X is topological vector space and f([a, b]) is connected for every segment [a, b] ? X. We show that if X is a hereditarily Baire space, Y is a metric space, f: X → Y is a Baire-one function and one of the following conditions holds:

1. X is a connected and locally connected space and f is a weakly Gibson function
2. X is an arcwise connected space and f is a Darboux function
3. X is a topological vector space and f is a segmentary connected function, then f has a connected graph.

     

Meromorphic functions that share a set with their derivatives

There exists a set S with three elements such that if a meromorphic function f, having at most finitely many simple poles, shares the set S CM with its derivative f^′, then f^′≡f.     

Meromorphic functions of finite order with maximal deficiency sum

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 55, pp. 84–95, 1991.