Inequalities play a fundamental role in Functional Analysis and it is widely recognized that finding them, especially sharp estimates, is an art. Together with the mathematical proofs the author also presents numerous applications to the calculus of variations and to many problems of quantum physics, in particular to atomic physics.

Commentaries.- I. Inequalities Related to Statistical Mechanics and Condensed Matter.- I.1 Theory of Ferromagnetism and the Ordering of Electronic Energy Levels.- I.2 Ordering Energy Levels of Interacting Spin Systems.- I.3 Entropy Inequalities (with H. Araki).- I.4 A Fundamental Property of Quantum-Mechanical Entropy.- I.5 Proof of the Strong Subadditivity of Quantum-Mechanical Entropy.- I.6 Some Convexity and Subadditivity Properties of Entropy.- I.7 A Refinement of Simon’s Correlation Inequality.- I.8 Two Theorems on the Hubbard Model.- I.9 Magnetic Properties of Some Itinerant-Electron Systems at T > 0.- II. Matrix Inequalities and Combinatorics.- II.1 Proofs of Some Conjectures on Permanents.- II.2 Concavity Properties and a Generating Function for Stirling Numbers.- II.3 Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture.- II.4 Some Operator Inequalities of the Schwarz Type.- II.5 Inequalities for Some Operator and Matrix Functions.- II.6 Positive Linear Maps Which Are Order Bounded on C* Subalgebras.- II. 7 Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration Inequalities.- II. 8 Sharp Uniform Convexity and Smoothness Inequalities for Trace Norms.- II.9 A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy.- III. Inequalities Related to the Stability of Matter.- III.1 Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities.- III.2 On Semi-Classical Bounds for Eigenvalues of Schrödinger Operators.- III.3 The Number of Bound States of One-Body Schrödinger Operators and the Weyl Problem.- III.4 Improved Lower Bound on the Indirect Coulomb Energy.- III.5 Density Functionals for Coulomb Systems.- III.6 On Characteristic Exponents inTurbulence.- III.7 Baryon Mass Inequalities in Quark Models.- III.8 Kinetic Energy Bounds and Their Application to the Stability of Matter.- III.9 A Sharp Bound for an Eigenvalue Moment of the One-Dimensional Schrödinger Operator.- IV. Coherent States.- IV.1 The Classical Limit of Quantum Spin Systems.- IV.2 Proof of an Entropy Conjecture of Wehrl.- IV.3 Quantum Coherent Operators: A Generalization of Coherent States.- IV.4 Coherent States as a Tool for Obtaining Rigorous Bounds.- V. Brunn-Minkowski Inequality and Rearrangements.- V.1 A General Rearrangement Inequality for Multiple Integrals.- V.2 Some Inequalities for Gaussian Measures and the Long-Range Order of the One-Dimensional Plasma.- V.3 Best Constants in Young’s Inequality, Its Converse and Its Generalization to More than Three Functions.- V.4 On Extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, Including Inequalities for Log Concave Functions and with an Application to the Diffusion Equation.- V.5 Existence and Uniqueness of the Minimizing Solution of Choquard’s Nonlinear Equation.- V.6 Symmetric Decreasing Rearrangement Can Be Discontinuous.- V.7 The (Non) Continuity of Symmetric Decreasing Rearrangement.- V.8 On the Case of Equality in the Brunn-Minkowski Inequality for Capacity.- VI. General Analysis.- VI.1 An Lp Bound for the Riesz and Bessel Potentials of Orthonormal Functions.- VI.2 A Relation Between Pointwise Convergence of Functions and Convergence of Functionals.- VI.3 Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities.- VI.4 On the Lowest Eigenvalue of the Laplacian for the Intersection of Two Domains.- VI.5 Minimum Action Solutions of Some Vector Field Equations.- VI.6 Sobolev Inequalities with Remainder Terms (with H. Brezis).- VI.7 Gaussian KernelsHave Only Gaussian Maximizers.- VI.8 Integral Bounds for Radar Ambiguity Functions and Wigner Distributions.- VII. Inequalities Related to Harmonic Maps.- VII.1 Estimations d’énergie pour des applications de R3 à valeurs dans S2.- VII.2 Singularities of Energy Minimizing Maps from the Ball to the Sphere.- VII.3 Co-area, Liquid Crystals, and Minimal Surfaces.- VII.4 Counting Singularities in Liquid Crystals (with F. Almgren).- VII.5 Symmetry of the Ginzburg-Landau Minimizer in a Disc.- Publications of Elliott H. Lieb.

Essential tool when applying analysis to problems of practical importance Brings together a host of inequalities otherwise scattered in the literature and hard to find Includes supplementary material: sn.pub/extras

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