Inequalities

Inequalities arise as an essential component in various mathematical areas. Besides forming a highly important collection of tools, e.g. for proving analytic or stochastic theorems or for deriving error estimates in numerical mathematics, they constitute a challenging research field of their own. Inequalities also appear directly in mathematical models for applications in science, engineering, and economics.

This book concentrates on one- and multi-dimensional nonlinear integral and discrete Gronwall-Bellman type inequalities. It complements the author’s book on linear inequalities and serves as an essential tool for researchers interested in differential (ODE and PDE), difference, and integral equations.

Inequalities continue to play an essential role in mathematics. Perhaps, they form the last field comprehended and used by mathematicians in all areas of the discipline. Since the seminal work Inequalities (1934) by Hardy, Littlewood and Pólya, mathematicians have laboured to extend and sharpen their classical inequalities.

Aimed toward researchers, postgraduate students, and scientists in linear operator theory and mathematical inequalities, this self-contained monograph focuses on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. Students at the graduate level will learn some essentials that may be useful for reference in courses in functional analysis, operator theory, differential equations, and quantum computation, to name several. Chapter 1 presents fundamental facts about the numerical range and the numerical radius of bounded linear operators in Hilbert spaces.