Value Distribution Theory and Related Topics

The Nevanlinna theory of value distribution of meromorphic functions, one of the milestones of complex analysis during the last century, was c- ated to extend the classical results concerning the distribution of of entire functions to the more general setting of meromorphic functions. Later on, a similar reasoning has been applied to algebroid functions, subharmonic functions and meromorphic functions on Riemann surfaces as well as to - alytic functions of several complex variables, holomorphic and meromorphic mappings and to the theory of minimal surfaces. Moreover, several appli- tions of the theory have been exploited, including complex differential and functional equations, complex dynamics and Diophantine equations. The main emphasis of this collection is to direct attention to a number of recently developed novel ideas and generalizations that relate to the - velopment of value distribution theory and its applications. In particular, we mean a recent theory that replaces the conventional consideration of counting within a disc by an analysis of their geometric locations. Another such example is presented by the generalizations of the second main theorem to higher dimensional cases by using the jet theory. Moreover, s- ilar ideas apparently may be applied to several related areas as well, such as to partial differential equations and to differential geometry. Indeed, most of these applications go back to the problem of analyzing zeros of certain complex or real functions, meaning in fact to investigate level sets or level surfaces.

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The volume consists of a collection of articles on the value distribution theory and its applications, both in one and several variables. The applied parts include problems related to geometric function theory, linear operators of entire functions, differential and functional equations, uniqueness and interpolation. A unique feature of the book is an extensive research program by the first editor on the Gamma-lines approach to analysis. Some aspects in the book consider Diophantine type equations for meromorphic functions, offering new challenges to complex analysis. Audience: Researchers and postgraduate students in complex analysis, differential equations and algebraic geometry would find this book of interest.

Contenu
Geometric value distribution theory.- A New Program of Investigations in Analysis: Gamma-Lines Approaches.- On Level Sets of Quasiconformal Mappings.- Classical value distribution theory.- On the Unintegrated Nevanlinna Fundamental Inequality for Meromorphic Functions of Slow Growth.- On Some New Concept of Exceptional Values.- Maximum Modulus Points, Deviations and Spreads of Meromorphic Functions.- Composition Theorems, Multiplier Sequences and Complex Zero Decreasing Sequences.- Nevanlinna Theory in an Annulus.- On Strong Asymptotic Tracts of Functions Holomorphic in a Disk.- Complex differential and functional equations.- A New Trend in Complex Differential Equations: Quasimeromorphic Solutions.- On the Functional Equation P(F)=Q(G).- Value Distribution of the Higher Order Analogues of the First Painlevé Equation.- Some Further Results on the Functional Equation P(F)=Q(G).- Several variables theory.- Recent Topics in Uniqueness Problem for Meromorphic Mappings.- On Interpolation Problems in Cn.- Jet Bundles and its Applications in Value Distribution of Holomorphic Mappings.- Normal Families of Meromorphic Mappings of Several Complex Variables into the Complex Projective Space.