Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\Cal K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with $d\leq n+k$. If $f_1, f_2, \hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\Cal K(M)$ is a simple finite algebraic extension of the field $\Bbb C(f_1, f_2, \hdots, f_d)$ of rational functions of the $f_1, f_2, \hdots , f_d$. When $M$ has a projective embedding, there is an analogue of Chow's theorem, and $\Cal K(M)$ is isomorphic to the field $\Cal R(Y)$ of rational functions on an irreducible projective algebraic variety $Y$, and $M$ has a $CR$ embedding in $\roman{reg} Y$. The equivalence between algebraic dependence and analytic dependence fails when condition $E$ is dropped.
A two-parameter characteristic of functions meromorphic on annuli is introduced and an extension of the Nevanlinna value distribution theory for such functions is proposed.
Let M be a smooth CR manifold of CR dimension n and CR codimension k, which is not compact, but has the local extension property E. We introduce the notion of "elementary pseudoconcavity" for M, which extends to CR manifolds the concept of a "pseudoconcave" complex manifold. This notion is then used to obtain generalizations, to the noncompact case, of the results of our previous paper about algebraic dependence, transcendence degree and related matters for the field K(M) of CR meromorphic functions on M.
In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines the topological type of the generic real meromorphic function and that the set of all parks is in $1-1$-correspondence with the set of all connected components in the space of generic real meromorphic functions. For any of the above components, we introduce and calculate the corresponding Hurwitz number. Finally as a consequence of our results we determine t…
We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of polynomial functions.
We present some results on two meromorphic functions from S to the Riemann sphere sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane.
A univalent meromorphic function defined on $Δ:= \{z \in \mathbb{C}: 1<|z|<\infty \}$ with univalent inverse defined on $Δ$ is bi-univalent meromorphic in $Δ$. For certain subclasses of meromorphic bi-univalent functions, estimates on the initial coefficients are obtained.
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.
We describe bounded, holomorphic functions on the complex 2-disc, that admit meromorphic extension to a larger 2-disc. This solves a conjecture of Bickel, Knese, Pascoe and Sola. The key technical ingredient is an old theorem of Zariski about integrally closed ideals in 2-dimensional regular rings. v2: Remark 15 is added.
In the work, we focus on a conjecture due to Z.X. Chen and H.X. Yi[1] which is concerning the uniqueness problem of meromorphic functions share three distinct values with their difference operators. We prove that the conjecture is right for meromorphic function of finite order. Meanwhile, a result of J. Zhang and L.W. Liao[10] is generalized from entire functions to meromorphic functions.
We study ordinary differential equations in the complex domain given by meromorphic vector fields on Kähler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in particular, if it has an entire one) then, up to a bimeromorphic transformation, either the vector field is holomorphic or it preserves a fibration.
In this paper, we study primeness and pseudo primeness of p-adic meromorphic functions. We also consider left (resp. right ) primeness of these functions. We give, in particular, sufficient conditions for a meromorphic function to satisfy such properties. Finally, we consider the problem of permutability of entire functions.
We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P_1, i.e., which have bounded spherical derivative.
In the present paper, we introduced the extended bicomplex plane $\bar{\mathbb{T}}$, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about the convergence of the sequences of bicomplex meromorphic functions. Hence the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bico…
Thirty research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.
A result is proved concerning meromorphic functions of finite order in the plane such that all but finitely many zeros of the second derivative are zeros of the first derivative.
In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang~\cite{WF} and J. Qui, T. Zhu ~\cite{QZ}.
This paper is devoted to establish sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extend some results for entire functions to meromorphic functions. Actually, we shall mainly discuss non-existence of unbounded wandering domains of a meromorphic function. The case for a composition of finitely many meromorphic function with at least one of them being transcendental can be also investigated in the argument of this paper.
For any connected component $H_0$ of the space of real meromorphic functions we build a compactification $N(H_0)$ of the space $H_0$. Then we express the Euler characteristics of the spaces $H_0$ and $N(H_0)$ in terms of topological invariants of functions from $H_0$.
We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng Problem.
We study the convergence of sequences of type I and type II Hermite-Padé approximants for certain systems of meromorphic functions made up of rational modifications of Nikishin systems of functions.
We determine all pairs $(f,g)$ of meromorphic functions that share four pairs of values $(a_ν,b_ν)$, $1\leν\le 4$, and a fifth pair $(a_5,b_5)$ under some mild additional condition.
In 1949 Siegel gave an example of a complex two-torus with no nonconstant meromorphic functions. In 1964 Kodaira showed that compact complex surfaces with no nonconstant meromorphic must be of the following three types: tori, Hopf type surfaces with first Betti number equal to one, and K3 surfaces. In this paper we show that surfaces of these three types have a dense set of surfaces in their natural moduli spaces with no nonconstant meromorphic functions.
The main purpose of this paper is to derive some subordination and superordination results involving certain of integral operator for meromorphic univalent functions in the punctured open unit disk. Several sandwich-type results are also obtained.
In this paper we wish to prove some results related to the growth rates of entire and meromorphic functions on the basis of relative (p,q) th order and relative (p,q) th type of a meromorphic function with respect to an entire function for any two positive integers p and q.
The object of this paper is studying some properties of meromorphic functions which satisfy in the condition \[Re(zf(z)) > α|z^2f'(z)+zf(z)| .\] Parallel results for some related classes are also obtained.
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.
We study meromorphic functions in a strip almost periodic with respect to the spherical metric. Then we get a complete description of zeros and poles for this class of functions, find a condition for a meromorphic almost periodic function to be a quotient of two holomorphic almost periodic function, find a necessary and suficient condition for sum and product of almost periodic meromorphic functions to be almost periodic.
We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.
The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated. The $φ$-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order $ρ_{\log}(f)$ and the classical order $ρ(f)$ of a meromorphic function $f$.
Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple exept finitely many and T(r,h)=o{T(r,f)} as r tends to infinity, then f'=h has infinitely many solutions (including poles).
In this paper we prove some normality criteria for a family of meromorphic functions, which involves the zeros of certain differential polynomials generated by the members of the family.
In this paper we extend the concept of bi-univalent to the class of meromorphic functions. We propose to investigate the coefficient estimates for two classes of meromorphic bi-univalent functions. Also, we find estimates on the coefficients |b0| and |b1| for functions in these new classes. Some interesting remarks and applications of the results presented here are also discussed.
Orders and types of entire and meromorphic functions have been actively investigated by many authors. In the present paper, we aim at investigating some basic properties in connection with sum and product of relative $(p,q)$-$φ$ order, relative $(p,q)$-$φ$ type, and relative $(p,q)$-$φ$ weak type of meromorphic functions with respect to entire functions.
This paper is devoted to investigate the singular directions of mero- morphic functions in some angular domains. We will confirm the existence of Hayman T directions in some angular domains. This is a continuous work of Yang [Yang L., Borel directions of meromorphic functions in an angular domain, Science in China, Math. Series(I)(1979), 149-163.] and Zheng [Zheng, J.H., Value Distribution of Meromorphic Functions, preprint.].
The paper determines all meromorphic functions with finitely many zeros in the plane having the property that a linear differential polynomial in the function, of order at least 3 and with rational functions as coefficients, also has finitely many zeros.
We give several unequivalent notions of convergency of meromorphic functions and more generally meromorphic mappings (strong, weak, $Γ$-convergency and some others). Relations between them are investigated. A version of Rouche theorem for the strongly converging sequences of meromorphic mappings is given. Further we investigate the sets of normality of families of meromorphic mappings, connections with extension properties of those mappings, some apriori estimates for the volume. Finally we apply the technique developped to the study of Fatou sets of meromorphic endomorphisms of Kähler surface…
We study the convergence of type I Hermite-Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.
In this note it is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.
For a germ of a meromorphic function f=P/Q, we offer notions of the monodromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators.
We prove the rank one case of Skolem's Conjecture on the exponential local-global principle for algebraic functions and discuss its analog for meromorphic functions.
By using Nevanlinna theory, we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family.
In this paper, we investigate the value distribution for linear q-difference polynomials of transcendental meromorphic functions of zero order which improves the results of Xu, Liu and Cao (\cite{Xu & Liu & Cao & 2015}). We also investigate the uniqueness of zero order meromorphic function with its q-difference operator sharing two sets with finite weight. Some examples have been exhibited which are relevant to the content of the paper.
In this paper, we prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share three sets. The obtained results improve some recent existing results.
Every nonconstant meromorphic function in the plane univalently covers spherical discs of radii arbitrarily close to arctan(sqrt 8) ~ 70^\circ 32'. If in addition all critical points of the function are multiple, then a similar statement holds with pi/2. These constants are the best possible. The proof is based on the consideration of negatively curved singular surfaces associated with meromorphic functions.
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.
