Inequalities are useful in all fields of Mathematics. The aim of this problem-oriented book is to present elementary techniques in the theory of inequalities. The readers will meet classical theorems including Schur’s inequality, Muirhead’s theorem, the Cauchy-Schwarz inequality, the Power Mean inequality, the AM-GM inequality, and Hölder’s theorem. I would greatly appreciate hearing about comm...

In classical coding, a single quantum state is encoded into classical information. Decoding this classical information in order to regain the original quantum state is known to be impossible. However, one can attempt to construct a state which comes as close as possible. We give bounds on the smallest possible trace distance between the original and the decoded state which can be reached. We gi...

We obtain an inequality complementary to the Cauchy-Schwarz inequality in Hilbert space. The inequalities involving first three powers of a self-adjoint operator are derived. The inequalities include the bounds for the third central moment, as a special case. It is shown that an upper bound for the spectral radius of a matrix is a root of a particular cubic equation, provided all eigenvalues ar...