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Front Matter |
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Abstract
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A New Program of Investigations in Analysis: Gamma-Lines Approaches |
G. Barsegian |
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Abstract
A new program of mathematical studies primarily based on the theory of Gamma-lines and ideas of the Nevanlinna value distribution theory is presented. This program establishes new connections between a variety of mathematical fields: real and complex analysis, ordinary, partial and complex differential equations, differential geometry, real and complex algebraic geometry, and Hilbert’s topological problem 16. Preliminary results, related to some of the problems posed, are given. In addition, the usefulness of this program in applications will be discussed.
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On Level Sets of Quasiconformal Mappings |
G. A. Sukiasyan |
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Abstract
In the present article some analogs and generalizations of the tangent variation principle are given for quasiconformal and continuously differentiable mappings.
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On the Unintegrated Nevanlinna Fundamental Inequality for Meromorphic Functions of Slow Growth |
Angel Alonso,Arturo Fernández,Javier Pérez |
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Abstract
J. Miles proved that for a meromorphic function . and . values ., ..., ., the inequality ., holds for some constant ., for all large . in a set of positive lower logarithmic density. This inequality is in some sense stronger than the unintegrated Nevanlinna fundamental inequality .. However, it remains the question about the size of the constant .. In this work, the above mentioned inequality will be considered for functions of slow and regular growth, observing that in this case, which is a natural extension of the rational functions class, the constant . can be considerably reduced in relation with the numerical values suggested by Miles. We make use of a result of Barsegian which follows from some beautiful considerations around the main theorems of the Ahlfors theory of covering surfaces.
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On Some New Concept of Exceptional Values |
G. A. Barsegian,C. C. Yang |
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Abstract
We introduce a concept of . . for functions . meromorphic in the complex plane. This concept generalizes the classical concept of multiple points. By making use of the new concept, we are able to generalize some main conclusions of the Nevanlinna value distribution theory related to multiple .-.. In particular, it turns out that not only these multiple .-. are exceptional in the sense of deficiency but also those .-. . where . is sufficiently small.
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Maximum Modulus Points, Deviations and Spreads of Meromorphic Functions |
E. Ciechanowicz,I. I. Marchenko |
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Abstract
We consider the influence of the number of maximum modulus points over the spread and the magnitude of deviation of meromorphic functions.
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Composition Theorems, Multiplier Sequences and Complex Zero Decreasing Sequences |
Thomas Craven,George Csordas |
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Abstract
An important chapter in the theory of distribution of zeros of polynomials and transcendental entire functions pertains to the study of linear operators acting on entire functions. This article surveys some recent developments (as well as some classical results) involving some specific classes of linear operators called multiplier sequences and complex zero decreasing sequences. This expository article consists of four parts: Open problems and background information, Composition theorems (Section 2), Multiplier sequences and the Laguerre-Pólya class (Section 3) and Complex zero decreasing sequences (Section 4). A number of open problems and questions are also included.
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Nevanlinna Theory in an Annulus |
Risto Korhonen |
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Abstract
A concrete presentation of Nevanlinna theory in a domain . has been offered by Bieberbach. He applied Green’s formula to prove the first main theorem and the lemma of the logarithmic derivative for meromorphic functions outside a disc of radius .. Apart from this work, Nevanlinna theory outside a disc has been considered in the form of brief remarks only in various articles. The purpose of this paper is to collect these comments into a coherent presentation, and to generalize these results for functions meromorphic in an open annulus. We define annulus versions of the Nevanlinna functions allowing accumulation of poles also to the inner boundary, and prove analogues of Nevanlinna’s main theorems including the lemma of the logarithmic derivative. Instead of using Green’s formula, we base our reasoning on a theorem due to Valiron.
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On Strong Asymptotic Tracts of Functions Holomorphic in a Disk |
I. I. Marchenko,I. G. Nikolenko |
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Abstract
For entire functions of finite order the Ahlfors classical theorem about finiteness of the set of asymptotic values is well known. In 1999, the first author introduced the concept of the strong asymptotic value of entire functions and obtained an analogue of the Ahlfors theorem for distinct strong asymptotic spots of entire functions of infinite order [14]..In 2001 we introduced the concept of a strong asymptotic spot in a point . for functions holomorphic in the disk. For such asymptotic spots we obtained an analogue of the Ahlfors theorem for functions holomorphic in the disk..In the case of holomorphic functions of order . MacLane proved that the set of distinct asymptotic tracts corresponding to the point . is finite [12]. He also obtained an estimate for their quantity. We introduce the concept of a strong asymptotic tracts for functions holomorphic in a disk and obtain an analogue of the Ahlfors theorem for their quantity for functions holomorphic in a disk.
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A New Trend in Complex Differential Equations: Quasimeromorphic Solutions |
G. A. Barsegian,A. A. Sarkisian,C. C. Yang |
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Abstract
In complex differential equations, one usually studies analytic or meromorphic solutions. We start a new trend by considering quasimeromorphic solutions for generalized algebraic differential equations of the first order. In particular, the classical Goldberg result that any meromorphic solution of a first order algebraic differential equation must be of finite order will been extended here to .-. solutions of first order generalized algebraic differential equations.
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On the Functional Equation |
Huy Khoai Ha,C. C. Yang |
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Abstract
We prove that for a generic pair (.) of polynomials . of degree . and . of degree ., where . are satisfying some conditions, . for meromorphic functions . implies ., .. We also give another proof of the statement saying that a generic polynomial of degree at least 5 is a uniqueness polynomial for meromorphic functions.
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Value Distribution of the Higher Order Analogues of the First Painlevé Equation |
Yuzan He |
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Abstract
The higher order analogues of the first higher order Painlevé equation (.) arise from an exact reduction of the higher order .-. (.). In this paper we examine value distribution properties of the meromorphic solutions of (.).
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Some Further Results on the Functional Equation |
Chung-Chun Yang,Ping Li |
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Abstract
In this paper we study further the solvability, i.e. the existence of nonconstant meromorphic solutions ., of the functional equation ., where . are two polynomials in .[.]. Some criteria for such questions in terms of the degrees of . and . as well as their coefficients are established.
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Recent Topics in Uniqueness Problem for Meromorphic Mappings |
Yoshihiro Aihara |
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Abstract
In this article, we give a survey of recent development of uniqueness problem for meromorphic mappings. In particular, we give an overview of its applications to constructing problem of hyperbolic hypersurfaces in complex projective spaces. Furthermore, we give a review on some recent researches on unique range set for meromorphic functions of one complex variable.
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On Interpolation Problems in Cn |
Carlos A. Berenstein,Bao Qin Li |
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Abstract
We will consider under what conditions an analytic variety in . is an interpolating variety for weighted spaces of entire functions, which is one of fundamental problems in several complex variables. Interest in this area arises from connections and applications of such questions to other problems such as representation of solutions of partial differential equations, deconvolution, and the Nullstellensatz. We shall discuss some of recent results on this subject, with a special attention given to those by the authors and their collaborators.
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Jet Bundles and its Applications in Value Distribution of Holomorphic Mappings |
Pei-Chu Hu,Chung-Chun Yang |
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Abstract
In this paper, we have established the technique of the higher dimensional jets and applied the results to study value distribution of holomorphic mappings. As applications, we have also generalized the results of holomorphic curves obtained by Ochiai, Noguchi and Green-Griffiths to the higher dimensional cases.
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Normal Families of Meromorphic Mappings of Several Complex Variables into the Complex Projective Spa |
Zhen-Han Tu |
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Abstract
In this paper, we discuss normality criteria for families of holomorphic mappings and meromorphic mappings of several complex variables into the complex projective space related to Green’s and Nochka’s Picardtype theorems, and improve an earlier result of singular directions for holomorphic curves in the complex projective space. Some related topics will also be given here.
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