Operator Inequalities of the Jensen, Čebyšev and Grüss Type [electronic resource] /
Dragomir, Silvestru Sever.author.
The main aim of this book is to present recent results concerning inequalities of the Jensen, Čebyšev 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 Čebyšev, (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.
Description based upon print version of record.
Operator Inequalitiesof the Jensen, Cebyšev and Grüss Type; Abstract; Preface; Contents; Chapter 1 Functions of Selfadjoint Operators on Hilbert Spaces; 1.1 Introduction; 1.2 Bounded Selfadjoint Operators; 1.2.1 Operator Order; 1.3 Continuous Functions of Selfadjoint Operators; 1.3.1 Polynomials in a Bounded Operator; 1.3.2 Continuous Functions of Selfadjoint Operators; 1.4 Step Functions of Selfadjoint Operators; 1.5 The Spectral Decomposition of Selfadjoint Operators; 1.5.1 Operator Monotone and Operator Convex Functions; References; Chapter 2 Inequalities of the Jensen Type
2.1 Introduction2.2 Reverses of the Jensen Inequality; 2.2.1 An Operator Version of the Dragomir-Ionescu Inequality; 2.2.2 Further Reverses; 2.3 Some Slater Type Inequalities; 2.3.1 Slater Type Inequalities for Functions of Real Variables; 2.3.2 Some Slater Type Inequalities for Operators; 2.3.3 Further Reverses; 2.4 Other Inequalities for Convex Functions; 2.4.1 Some Inequalities for Two Operators; 2.5 Some Jensen Type Inequalities for Twice Differentiable Functions; 2.5.1 Jensen's Inequality for Twice Differentiable Functions; 2.6 Some Jensen's Type Inequalities for Log-Convex Functions
2.6.1 Preliminary Results2.6.2 Jensen's Inequality for Differentiable Log-ConvexFunctions; 2.6.3 More Inequalities for Differentiable Log-ConvexFunctions; 2.6.4 A Reverse Inequality; 2.7 Hermite-Hadamard's Type Inequalities; 2.7.1 Scalar Case; 2.7.2 Some Inequalities for Convex Functions; 2.8 Hermite-Hadamard's Type Inequalities for Operator Convex Functions; 2.8.1 Introduction; 2.8.2 Some Hermite-Hadamard's Type Inequalities; 2.8.3 Some Operator Quasi-linearity Properties; References; Chapter 3 Inequalities of the Cebyšev and Grüss Type; 3.1 Introduction; 3.2 Cebyšev's Inequality
3.2.1 Cebyšev's Inequality for Real Numbers3.2.2 A Version of the Cebyšev Inequality for One Operator; 3.2.3 Related Results for One Operator; 3.3 Grüss Inequality; 3.3.1 Some Elementary Inequalities of Grüss Type; 3.3.2 An Inequality of Grüss' Type for One Operator; 3.4 More Inequalities of Grüss Type; 3.4.1 Some Vectorial Grüss' Type Inequalities; 3.4.2 Some Inequalities of Grüss' Type for One Operator; 3.5 More Inequalities for the Cebyšev Functional; 3.5.1 A Refinement and Some Related Results; 3.6 Bounds for the Cebyšev Functional of Lipschitzian Functions
3.6.1 The Case of Lipschitzian Functions3.6.2 The Case of ( f,?)-Lipschitzian Functions; 3.7 Quasi-Grüss' Type Inequalities; 3.7.1 Introduction; 3.7.2 Vector Inequalities; 3.7.3 Applications for Grüss' Type Inequalities; 3.8 Two Operators Grüss' Type Inequalities; 3.8.1 Some Representation Results; 3.8.2 Bounds for f of Bounded Variation; 3.8.3 Bounds for f Lipschitzian; 3.8.4 Bounds for f Monotonic Non-decreasing; References
Includes bibliographical references.
The main aim of this book is to present recent results concerning inequalities of the Jensen, Čebyšev 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 Čebyšev, (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.
Operator theory.
Partial differential equations.
Probabilities.
Operator Theory.
Partial Differential Equations.
Probability Theory and Stochastic Processes.
SpringerBriefs in Mathematics