536 research outputs found
Further results on normal families of meromorphic functions concerning shared values
In this paper, we prove two normality criteria for families of some functions
concerning shared values, the results generalize those given by Hu and Meng.
Some examples are given to show the sharpness of our results.Comment: 9page
Normality criteria for a family of meromorphic functions with multiple zeros
In this article, we prove some normality criteria for a family of meromorphic
functions having zeros with some multiplicity. Our main result involves sharing
of a holomorphic function by certain differential polynomials. Our results
generalize some of the results of Fang and Zalcman and Chen et al to a great
extent
Uniqueness of Some Differential Polynomials of Meromorphic Functions
In this paper, we prove some uniqueness results which improve and generalize
several earlier works. Also, we prove a value distribution result concerning
which provides a partial answer to a question of Fang and Wang [A
note on the conjectures of Hayman, Mues and Gol'dberg, Comp. Methods, Funct.
Theory (2013)13, 533-543].Comment: 14 page
Normality criteria concerning composite meromorphic functions
In this paper, we prove normality criteria for families of meromorphic
functions involving sharing of a holomorphic function by a certain class of
differential polynomials. Results in this paper extends the works of different
authors carried out in recent years.Comment: 14 page
Some Normality Criteria
In this article we prove some normality criteria for a family of meromorphic
functions which involves sharing of a non-zero value by certain differential
monomials generated by the members of the family. These results generalizes
some of the results of Schwick.Comment: 16 Pages. Comments are welcome, Communicate
Zeros of differential polynomials in real meromorphic functions
We show that for a real transcendental meromorphic function f, the
differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros.
Similar results are obtained for differential polynomials f'f^m-1. We specially
investigate the case of meromorphic functions with finitely many poles. We show
by examples the precision of our results. One of our main tools is the Fatou
theorem from complex dynamics.Comment: 18 page
Bloch's principle
A heuristic principle attributed to A. Bloch says that a family of
holomorphic functions is likely to be normal if there is no nonconstant entire
functions with this property. We discuss this principle and survey recent
results that have been obtained in connection with it. We pay special attention
to properties related to exceptional values of derivatives and existence of
fixed points and periodic points, but we also discuss some other instances of
the principle.Comment: 30 page
Dynamics of Generalized Nevanlinna Functions
In the early 1980's, computers made it possible to observe that in complex
dynamics, one often sees dynamical behavior reflected in parameter space and
vice versa. This duality was first exploited by Douady, Hubbard and their
students in early work on rational maps. See \cite{DH,BH} for example. Here, we
continue to study these ideas in the realm of transcendental functions.
In \cite{KK1}, it was shown that for the tangent family, ,
the way the hyperbolic components meet at a point where the asymptotic value
eventually lands on infinity reflects the dynamic behavior of the functions at
infinity. In the first part of this paper we show that this duality extends to
a much more general class of transcendental meromorphic functions that we call
{\em generalized Nevanlinna functions} with the additional property that
infinity is not an asymptotic value. In particular, we show that in
"dynamically natural" one dimensional slices of parameter space, there are
"hyperbolic-like" components with a unique distinguished boundary point whose
dynamics reflect the behavior inside an asymptotic tract at infinity. Our main
result is that {\em every} parameter point in such a slice for which the
asymptotic value eventually lands on a pole is such a distinguished boundary
point.
In the second part of the paper, we apply this result to the families
, , to prove that all hyperbolic
components of period greater than are bounded.Comment: 31 pages, 3 figure
Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group
We prove Patterson's conjecture about the singularities of the Selberg zeta
function associated to a convex-cocompact, torsion free group acting on a
hyperbolic space.Comment: 63 pages, published versio
Meromorphic functions of one complex variable. A survey
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.Comment: 33 page
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