The main aim of this book is to present recent results concerning inequalities of the Jensen
?ebysev and Grüss type for continuous functions of bounded selfadjoint operators on complex
Hilbert spaces. In the introductory chapter the author portrays fundamental facts concerning
bounded selfadjoint operators on complex Hilbert spaces. The generalized Schwarz's inequality
for positive selfadjoint operators as well as some results for the spectrum of this class of
operators are presented. This text introduces the reader to the fundamental results for
polynomials in a linear operator continuous functions of selfadjoint operators as well as the
step functions of selfadjoint operators. The spectral decomposition for this class of operators
which play a central role in the rest of the book and its consequences are introduced. At the
end of the chapter some classical operator inequalities are presented as well. Recent new
results that deal with different aspects of the famous Jensen operator inequality are explored
through the second chapter. These include but are not limited to the operator version of the
Dragomir-Ionescu inequality the Slater type inequalities for operators and its inverses
Jensen's inequality for twice differentiable functions whose second derivatives satisfy some
upper and lower bound conditions and Jensen's type inequalities for log-convex functions.
Hermite-Hadamard's type inequalities for convex functions and the corresponding results for
operator convex functions are also presented. The ?ebysev (Chebyshev) inequality that compares
the integral discrete mean of the product with the product of the integral discrete means is
famous in the literature devoted to Mathematical Inequalities. The sister inequality due to
Grüss which provides error bounds for the magnitude of the difference between the integral mean
of the product and the product of the integral means has also attracted much interest since it
has been discovered in 1935 with more than 200 papers published so far. The last part of the
book is devoted to the operator versions of these famous results for continuous functions of
selfadjoint operators on complex Hilbert spaces. Various particular cases of interest and
related results are presented as well. This book is intended for use by both researchers in
various fields of Linear Operator Theory and Mathematical Inequalities domains which have
grown exponentially in the last decade as well as by postgraduate students and scientists
applying inequalities in their specific areas.hebyshev) inequality that compares the integral
discrete mean of the product with the product of the integral discrete means is famous in the
literature devoted to Mathematical Inequalities. The sister inequality due to Grüss which
provides error bounds for the magnitude of the difference between the integral mean of the
product and the product of the integral means has also attracted much interest since it has
been discovered in 1935 with more than 200 papers p
