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For more than a century, the study of various types of inequalities has been the focus of great attention by many researchers, interested both in the theory and its applications. In particular, there exists a very rich literature related to the well known Cebysev, Gruss, Trapezoid, Ostrowski, Hadamard and Jensen type inequalities. The present monograph is an attempt to organize recent progress related to the above inequalities, which we hope will widen the scope of their applications. The field to be covered is extremely wide and it is impossible to treat all of these here. The material included in the monograph is recent and hard to find in other books. It is accessible to any reader with a reasonable background in real analysis and an acquaintance with its related areas. All results are presented in an elementary way and the book could also serve as a textbook for an advanced graduate course. The book deserves a warm welcome to those who wish to learn the subject and it will also be most valuable as a source of reference in the field. It will be invaluable reading for mathematicians and engineers and also for graduate students, scientists and scholars wishing to keep abreast of this important area of research.

Inequalities (Mathematics). --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Inequalities (Mathematics) --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Mathematics. --- Functions of real variables. --- Real Functions. --- Real variables --- Functions of complex variables --- Math --- Science --- Processes, Infinite

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This book contains survey papers based on the lectures presented at the 3rd International Winter School “Modern Problems of Mathematics and Mechanics” held in January 2010 at the Belarusian State University, Minsk. These lectures are devoted to different problems of modern analysis and its applications. An extended presentation of modern problems of applied analysis will enable the reader to get familiar with new approaches of mostly interdisciplinary character. The results discussed are application oriented and present new insight into applied problems of growing importance such as applications to composite materials, anomalous diffusion, and fluid dynamics.

Mathematical analysis. --- Mathematical analysis --- Number theory --- Differential equations --- Functions of complex variables --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Calculus --- Applied Mathematics --- Mathematics. --- Math --- 517.1 Mathematical analysis --- Functions of complex variables. --- Partial differential equations. --- Functions of real variables. --- Number theory. --- Functions of a Complex Variable. --- Partial Differential Equations. --- Number Theory. --- Real Functions. --- Science --- Differential equations, partial. --- Number study --- Numbers, Theory of --- Algebra --- Partial differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Real variables

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This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.

Functions of complex variables --- 517.53 --- 517.53 Functions of a complex variable --- Functions of a complex variable --- Complex variables --- Elliptic functions --- Functions of real variables --- Graphic methods --- Graphic methods. --- Functions of complex variables. --- Functions, special. --- Functions of a Complex Variable. --- Special Functions. --- Special functions --- Mathematical analysis --- Special functions. --- Fonctions d'une variable complexe --- Methodes graphiques

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Functions of complex variables --- Mathematical analysis --- Fonctions d'une variable complexe --- Analyse mathématique --- Functions of complex variables. --- Mathematical analysis. --- Advanced calculus --- Analysis (Mathematics) --- Algebra --- Complex variables --- Elliptic functions --- Functions of real variables --- 517.1 Mathematical analysis --- Analyse mathématique --- 517.1. --- 517.1

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This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university education the content corresponds to a beginning graduate course. The book presents the fundamental results and methods of complex analysis and applies them to a study of elementary and non-elementary functions (elliptic functions, Gamma- and Zeta function including a proof of the prime number theorem …) and – a new feature in this context! – to exhibiting basic facts in the theory of several complex variables. Part of the book is a translation of the authors’ German text “Einführung in die komplexe Analysis”; some material was added from the by now almost “classical” text “Funktionentheorie” written by the authors, and a few paragraphs were newly written for special use in a master’s programme. Content Analysis in the complex plane - The fundamental theorems of complex analysis - Functions on the plane and on the sphere - Integral formulas, residues and applications - Non-elementary functions - Meromorphic functions of several variables - Holomorphic maps: Geometric aspects Readership Advanced undergraduates (bachelor students) and beginning graduate students (master's programme) Lecturers in mathematics About the authors Professor Dr. Ingo Lieb, Department of Mathematics, University of Bonn Professor Dr. Wolfgang Fischer, Department of Mathematics, University of Bremen.

Functions of complex variables. --- Mathematical analysis. --- Engineering & Applied Sciences --- Applied Mathematics --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Mathematics. --- Analysis (Mathematics). --- Analysis. --- Elliptic functions --- Functions of real variables --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic