CoSy seminar: "100 years celebration of the Hardy inequality- a new convexity proof and its consequences"

Everyone is welcome and the first 40 people to register will be treated to a free lunch sandwich. If you do not want lunch, you are still welcome to join.

Register for a lunch sandwich (before Sunday 7 December).

Abstract: First I shortly discuss the remarkable prehistory and history in connection to that G.H. Hardy, after around 10 years of research, published his most famous inequality, see [1]. I also present the early most popular extentions/variants (e.g. discrete, dual, powerweighted, Carleman, Polya-Knopp´s inequalities). It is nowadays well-known that convexity (Jensen´s inequality) more or less directly implies most of the classical inequalities. See e.g. my new book [2] and the references therein, but also this idea was known by Hardy himself,see the Hardy-Littlewood-Polya book. But what Hardy did not discover was that also his own inequality follows directly by using convexity (after a ”clever” substititution). In this talk, I present this two lines (Uppsala University related) proof, which gives an even more general inequality I call ”the fundamental form of Hardy´s inequality”, which, in particular, covers all variants mentioned above via elementary substitutions and limit procedures. After that I present some consequences of this new convexity approach, see [3], where e,g, the Persson -Samko equivalence theorem was proved and the natural question ”What should have happened if Hardy discovered this?” was raised.

Next I present another recently discovered surprising fact, namely that convexity (Jensen´s inequality) even can be used to prove a completely new type of generalization of Hölder´s inequality namely to a so called CONTINUOUS form (involving continously many functions, instead of 2 or finite many as in the classical situation). See my new book [4], where also many more such extensions of classical inequalities are proved and applied (e.g. to create a reaL interpolation theory between infinite many Banach spaces instead of two as in the classical case). Finally, I present some, in my opinion, remarkable results in the further developments of Hardy type inequalities as presented in my book [5]. Concerning the fact that this fascinating area is still developing in a remarkable way I also refer to [6]. Concerning important applications in the spirit of CIM I also refer to recently presented PhD theses by Harpal Singh, Kristoffer Tangrand and Markos Yimer.

[1] Kufner, A., Maligranda, L., Persson, L. E.: The prehistory of the Hardy inequality. Amer. Math. Monthly 113(8), 715 – 732 (2006).

[2] Niculescu, C., Persson, L. E.: Convex Functions and their Applications, A Contemporary Approach. Third Edition, CMS Books in Math, Springer, Cham, (2025).

[3] Persson, L. E., Samko, N.: What should have happened if Hardy had discovered this ?. J. Inequal. Appl. 2012, 2012:29, 11 pp.

[4] Nikolova, L., Persson, L.E., and Varosanec, S.: Continuous Versions of Some Classical Inequalities, Birkhäuser/Springer, (2025).

[5] Kufner, A., Persson, L.E., Samko, N.: Weighted Inequalities of Hardy Type. Second Edition, World Scientific Publication, New Jersey, (2017).

[6] Persson, L. E., Samko, N.: On Hardy-type inequalities as an intellectual adventure for 100 years. J. Math. Sci. 280, 180 – 197 (2024).

This is a lecture in the seminar series held by CIM (Centre for Interdisciplinary Mathematics).