This book is a self-contained advanced monograph on inequalities involving the numerical radius of bounded linear operators acting on complex Hilbert spaces. The study of numerical range and numerical radius has a long and distinguished history starting from the Rayleigh quotients used in the 19th century to nowadays applications in quantum information theory and quantum computing.

This monograph is intended for use by both researchers and graduate students of mathematics, physics, and engineering who have a basic background in functional analysis and operator theory. The book provides several challenging problems and detailed arguments for the majority of the results. Each chapter ends with some notes about historical views or further extensions of the topics. It contains a bibliography of about 180 items, so it can be used as a reference book including many classical and modern numerical radius inequalities.

P. Bhunia: He is a young and dynamic research scholar working in the Department of Mathematics, Jadavpur University, Kolkata, India, under the mentorship of Prof Kallol Paul. He is actively involved in research in the area of numerical radius inequalities. In his very short span of a research career, he has published several research papers in international journals of high reputation, which indicates his bright future in the coming days.

S. S. Dragomir: He is Professor and Chair of Mathematical Inequalities at Victoria University, Melbourne, Australia, and Honorary Professor at DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa. His research concerns Mathematical Inequalities ranging from real numbers and integrals to Hilbert space operators and Hermitian *-Algebras. He is also Chair of the International Research Group in Mathematical Inequalities and Applications and Editor-in-Chief of the Australia Journal of Mathematical Inequalities and Applications.

M. S. Moslehian: He is a Professor of Mathematics at the Ferdowsi University of Mashhad, Mashhad, Iran. His research concerns operator algebras, Hilbert C*-modules, in particular, operator and norm inequalities. He was a Senior Associate in ICTP (Italy) and a visiting professor at several universities in England, Sweden, China, and Japan. He is the editor-in-chief of the journals "Banach J. Math. Anal.", "Ann. Funct. Anal.", and "Adv. Oper. Theory" being published by Birkhäuser/Springer.

K. Paul: He is a Professor of Mathematics in the Department of Mathematics, Jadavpur University, Kolkata, India. He did his B.Sc. (Math.) Major from Gauhati University and M.Sc. from Jadavpur University, securing 1st Class 1st position in both examinations. His area of research is Operator Theory and Functional Analysis. His current area of teaching includes but is not limited to Real Analysis, Metric Spaces, Topology, and Operator Theory. Presently, he is actively involved in research in areas involving numerical radius inequalities, Birkhoff-James orthogonality, and their applications.