The main aim of this book is to present several results related to functions of unitary operators on complex Hilbert spaces obtained, by the author in a sequence of recent research papers. The fundamental tools to obtain these results are provided by some new Riemann-Stieltjes integral inequalities of continuous integrands on the complex unit circle and integrators of bounded variation. Features All the results presented are completely proved and the original references where they have been firstly obtained are mentioned Intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, as well as by postgraduate students and scientists applying inequalities in their specific areas Provides new emphasis to mathematical inequalities, approximation theory and numerical analysis in a simple, friendly and well-digested manner.About the AuthorSilvestru Sever Dragomir is Professor and Chair of Mathematical Inequalities at the College of Engineering & Science, Victoria University, Melbourne, Australia. He is the author of many research papers and several books on Mathematical Inequalities and their Applications. He also chairs the international Research Group in Mathematical Inequalities and Applications (RGMIA). For details, see https://rgmia.org/index.php.