Gamma-Lines: On the Geometry of Real and Complex Functions

Preface. Tangent Variation Principle. Satellite Principles. Modification of Length-area Principle. Tangent Variation Principle. Estimates for collections of Gamma-Lines. Estimates of lengths of Gamma-Lines for angular-quasiconformal mappings. Remarks on applying of estimates of L (D, Gamma). Nevanlinna and Ahilfors Theories. Additions. Basic concepts and outcomes of Nevanlinna Value Distribution theory and Ahlfors theory of covering surfaces. Geometric deficient values. On some additions to L. Ahlfor`s theory of covering surfaces. Bounds of some integrals. Gamma-Lines Approach in the Theory of Meromorphic Functions. Principle of closeness of sufficiently large sets of Alpha-points of meromorphic functions. Integrated Version of the Principle. Connections with known classes of functions. Distribution of Gamma-Lines for Functions Meromorphic in C. Applications. The main results on distribution of Gamma-Lines. \"Wingdings\" of Gamma-Lines. Average lengths of Gamma-Lines along concentric circles and the deficient values. Distribution of Gamma-Lines and value distribution of subclasses of modules and real parts of mermorphic functions. The number of Gamma-Lines crossing rings. Distribution of Gelfond points. Nevalinna`s dream-description of transcendental ramification of Riemann surfaces. The proximity property of Alpha-points of meromorphic functions. A proof of the proximity property of Alpha-points based only on investigation of Gamma-Lines. Some Applied Problems. Gamma-Lines in Physics. On the cross road of value distribution, Gamma-Lines, free boundary theories and applied mathematics. \"Pointmaps\" of physical processes and Alpha-points of general classes of functions Principles. Nevanlinna and Ahilfors Theories. Additions. Gamma-Lines Approach in the Theory of Meromorphic Functions. Distribution of Gamma-Lines for Functions Mermorphic in C. Applications. Some Applied Problems.