Abstract
The property of isotonicity of a continuous convex function on the positive cone is characterized via subdifferentials. This is used to illustrate a new generalization of the Hardy–Littlewood–Pólya inequality of majorization to the case of functions of a vector variable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, S., Persson, L.-E., Pečarić, J., Varošanec, S.: General inequalities via isotonic subadditive functionals. Math. Inequal. Appl. 10, 15–28 (2007)
Amann, H.: Multiple positive fixed points of asymptotically linear maps. J. Func. Anal. 17, 174–213 (1974)
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SlAM Rev. 18, 620–709 (1976)
Andô, T.: On fundamental properties of a Banach space with a cone. Pac. J. Math. 12, 1163–1169 (1962)
Bayart, F., Matheron, É.: Dynamics of linear operators. Cambridge Tracts in Mathematics 179. Cambridge University Press, Cambridge (2009)
Bogachev, S.: Measure Theory, vol. 2. Springer, Berlin (2007)
Borwein, J.M., Vanderwerff, J.: Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and Its Applications 109. Cambridge University Press, Cambridge (2010)
Buskes, G.: The Hahn–Banach theorem surveyed. Dissertationes Math. (Rozprawy Mat.) 327, 1–49 (1993)
Choquet, G.: Theory of capacities. Annales de l’ Institut Fourier 5, 131–295 (1954)
Davies, E.B.: The structure and ideal theory of the pre-dual of a Banach lattice. Trans. Am. Math. Soc. 131, 544–555 (1968)
Dellacherie, C.l.: Quelques commentaires sur les prolongements de capacités. Séminaire Probabilités V, Strasbourg, Lecture Notes in Math. vol. 191. Springer, Berlin and New York (1970)
Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic Publisher, Dordrecht (1994)
Feldman, M.M.: Sublinear operators defined on a cone. Siber. Math. J. 16, 1005–1015 (1975)
Gal, S.G.: Uniform and pointwise quantitative approximation by Kantorovich–Choquet type integral operators with respect to monotone and submodular set functions. Mediterr. J. Math. 14, 205–216 (2017)
Gal, S.G., Niculescu, C.P.: A Nonlinear extension of Korovkin’s theorem. Preprint arXiv:2003.00002 February 27, (2020)
Gal, S.G., Niculescu, C.P.: A note on the Choquet type operators. Preprint arXiv:2003.1430 April 1, (2020)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Hörmander, L.: Notions of Convexity. Reprint of the 1994 edition. Modern Birkhäuser Classics, Birkhäuser, Boston (2007)
Ioffe, A.D., Levin, V.L.: Subdifferentials of convex functions. Tr. Mosk. Mat. Obs. 26, 3–73 (1972)
Kantorovich, L.V.: On semi-ordered linear spaces and their applications to the theory of linear operators. Dokl. Akad. Nauk 4, 11–14 (1935)
Kusraev, A.G.: Dominated Operators. Springer, Berlin (2000)
Kusraev, A.G., Kutateladze S.S.: Subdifferentials: Theory and Applications. Mathematics and its Applications, vol. 323, Kluwer Academic Publishers Group, Dordrecht (1995)
Kutateladze, S.S.: Convex Operators. Russ. Math. Surveys 34(1), l8l–214 (1979)
Levin, V.L.: Subdifferentials of convex mappings and composite functions. Siber. Math. J. 13, 903–909 (1972)
Linke, YuE: On support sets of sublinear operators. Dokl. Akad. Nauk 207(3), 531–533 (1972)
Mesiar, R., Li, J., Pap, E.: The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46(6), 1098–1107 (2010)
Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)
Namioka, I.: Partially ordered linear topological spaces. Mem. Am. Math. Soc, No. 24 (1957)
Niculescu, C.P., Persson, L.-E.: Convex Functions and their Applications. A Contemporary Approach, 2nd Ed., CMS Books in Mathematics vol. 23, Springer, New York (2018)
Olteanu, O.: Convexité et prolongement d’opérateurs linéaires. C. R. Acad. Sci. Paris, Série A 286, 511–514 (1978)
Olteanu, O.: Théorèmes de prolongement d’opérateurs linéaires. Rev. Roumaine Math. Pures Appl. 28(10), 953–983 (1983)
Phelps, R.R.: Convex Functions, Monotone Operators, and Differentiability, Second Edition. Lecture Notes in Math. No. 1364, Springer, Berlin (1993)
Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series 28, Princeton University Press, Princeton, New Jersey (1970)
Rubinov, A.M.: Sublinear functionals defined on a cone. Siber. Math. J. 11, 327–336 (1970)
Rubinov, A.M.: Sublinear operators and their applications. Russ. Math. Surveys 32(4), 115–175 (1977)
Schaefer, H.H., Wolff, M.P.: Topological Vector Spaces. Graduate Texts in Mathematics 3, Second Edition. Springer, New York (1999)
Simon, B.: Convexity. An Analytic Viewpoint. Cambridge Tracts in Mathematics 187, Cambridge University Press, Cambridge (2011)
Valadier, M.: Sous-différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné. Math. Scand. 30, 65–74 (1972)
Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, New York (2009)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., River Edge, NJ (2002)
Zowe, J.: Sandwich theorems for convex operators with values in an ordered vector space. J. Math. Anal. Appl. 66, 282–296 (1978)
Acknowledgements
The authors would like to thank Professor Stefan Cobzas and Professor Gabriel Prajitura for several improvements on the original version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Niculescu, C.P., Olteanu, O. From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory. RACSAM 114, 171 (2020). https://doi.org/10.1007/s13398-020-00905-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00905-4
Keywords
- Order complete vector space
- Ordered Banach space
- Isotone function
- Convex function
- Subdifferential
- Majorization theory