Inequalities of averages; Hölder, Minkowski and Jensen inequalities; Stolarsky averages and Palés inequality; Khintchine theorem. Rearrangement inequalities and applications: Riesz, Chebyshev, Hardy-Littlewood-Pólya inequalities. Equivalence of summations, Schur convexity; symmetric functions and Muirhead’s theorem. Hilbert and Hardy inequalities; Wirtinger inequality; Steiner symmetrization; Hardy-Littlewood-Sobolev inequalities and applications; Brunn-Minkowski inequality and consequences: isoperimetric and Prekopa-Leindler inequalities.

References:
1. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, 2nd Edition, 1952.
2. D. J, H. Garling, Inequalities: A journey into Linear Analysis, Cambridge University Press, 2007.
3. K. Böröczky, A. Figalli and J. P. Ramos, Isoperimetric inequalities, Brunn-Minkowski theory and Minkowski-type Monge-Ampère equations on the sphere, Lecture notes to be published by EMS Press (preprint).

Note: This course is offered as a master’s course. In the doctorate, it has additional requirements.

 

* Basic syllabus. The teacher has the autonomy to make any changes.