Variations and Generalizations of Bohr′s Inequality

AbstractIn this paper we provide an account of various results that have been obtained concerning Bohr′s inequality with emphasis on several generalizations

On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure

Some General Families of Generating Functions for the Laguerre Polynomials

AbstractThe object of the present paper is to develop rather systematically some general families of bilinear, bilateral, or mixed multilateral generating functions for the classical Laguerre polynomials. Numerous straightforward consequences of some of the results considered here frequently appear in the literature, especially from the viewpoint of Lie groups and Lie algebras. It is also pointed out how the main generating functions can be suitably applied to derive numerous further results involving Laguerre polynomials and various other related polynomials

Preface

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