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This book presents the new fascinating area of continuous inequalities. It was recently discovered that several of the classical inequalities can be generalized and given in a more general continuous/family form. The book states, proves and discusses a number of classical inequalities in such continuous/family forms. Moreover, since many of the classical inequalities hold also in a refined form, it is shown that such refinements can be proven in the more general continuous/family frame.
Written in a pedagogical and reader-friendly way, the book gives clear explanations and examples on how this technique works. The presented interplay between classical theory of inequalities and these newer continuous/family forms, including some corresponding open questions, will appeal to a broad audience of mathematicians and serve as a source of inspiration for further research.
Ludmila Nikolova has dedicated most of her career to the field of Mathematical and Functional Analysis, progressing from a lecturer to a full professor at the Department of Mathematics and Informatics at Sofia University. Her scientific work began with interpolation of infinite families of Banach spaces, first with the complex method and later expanded to also include real interpolation. Inequalities is a field where she works actively too. Concerning her scientific career dedicated to interpolation theory and/or inequalities, a very fruitful collaboration first began with the second-named author and later also with the third-named author. Over time, this collaboration between the three authors has deepened and, in particular, inspired the writing of this book.
Lars-Erik Persson is Professor in mathematics at UiT The Arctic University of Norway, Senior Professor at Karlstad University, Sweden, and Professor Emeritus at Uppsala University, Sweden. His research interests are Inequalities, Convexity Theory, Fourier Analysis, Interpolation Theory, Homogenization Theory and Engineering Mathematics. He is a co-author of more than 300 journal papers, 10 research books, 5 teaching books on the university level and one teaching book on the pre-university level. He has been a supervisor of more than 70 students with PhD exams and received several awards for research, teaching and supervision. He has been President of the Swedish Mathematical Society.
Sanja Varosanec is a full professor at the Department of Mathematics, Faculty of Science of the University in Zagreb. Her research interests are mainly in Mathematical Analysis, particularly in the theory of inequalities. She is a co-author of two monographs: the first of them is focused on the Steffensen inequality and related results while the second is devoted to the general linear inequalities. Besides her scientific activities, she is deeply involved in mathematics teacher education. As a result of those efforts, she has published more than ten mathematical school textbooks and has written more than 40 professional papers for mathematics teachers.
- 1. Continuous Forms of Classical Inequalities.- 2. Refinements of Continuous Forms of Inequalities.- 3. Refinements of Inequalities via Strong Convexity and Superquadracity.- 4. Functionals Associated with Continuous Forms of Inequalities.- 5. Some Classical Inequalities Involving Banach Lattice Norms.
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