Skip to main content
SpringerLink
Log in

Functions with Positive Differences on Convex Cones

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We analyze the role played by functions with positive differences defined on convex cones. In particular, we study functions that satisfy linear functional inequalities that extend the three-variable Hornich-Hlawka functional inequality, \(f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left( x+y+z\right) \ge f\left( x+y\right) +f\left( y+z\right) +f\left( z+x\right) +f(0),\) especially to the case of n variables. Beyond the classical setting, we present extensions to the case of positive operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Amann, H.: Multiple positive fixed points of asymptotically linear maps. J. Funct. Anal. 17, 174–213 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bennett, G.: Some forms of majorization. Houst. J. Math. 36, 1037–1066 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Berndt, W., Sra, S.: Hlawka-Popoviciu inequalities on positive definite tensors. Linear Algebra Appl. 486, 317–327 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernstein, S.: Sur les fonctions absolument monotones. Acta Math. 52, 1–66 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhatia, R.: Matrix Analysis. Springer (1997)

  6. Bhatia, R.: Positive Definite Matrices. Princeton University Press (2007)

  7. Boas, R.P., Jr., Widder, D.V.: Functions with positive differences. Duke Math. J. 7, 496–503 (1940)

    MathSciNet  MATH  Google Scholar 

  8. Choquet, G.: Deux exemples classiques de représentation intégrale. Enseign. Math. 15, 63–75 (1969)

    MathSciNet  MATH  Google Scholar 

  9. Del Moral, P., Niclas, A.: A Taylor expansion of the square root matrix function. J. Math. Anal. Appl. 465, 259–266 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fréchet, M.: Sur la definition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert. Ann. Math. 36, 705–718 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gal, S.G.: Shape Preserving Approximation by Real and Complex Polynomials. Birkhäuser, Boston (2008)

    Book  MATH  Google Scholar 

  12. Gal, S.G., Niculescu, C.P.: A new look at Popoviciu’s concept of convexity for functions of two variables. J. Math. Anal. Appl. 479, 903–925 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hopf, E.: Über die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften. Dissertation, Univ. Berlin (1926)

  14. Hornich, H.: Eine Ungleichung für Vektorlängen. Math. Z. 48, 268–274 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  15. Komisarski, A., Rajba, T.: On the integral representation and the Raşa, Jensen and Hermite–Hadamard adamard inequalities for Box–Convex functions. Results Math. 78, 1–30 (2023)

    Article  MATH  Google Scholar 

  16. Kozhasov, K., Michalek, M., Sturmfels, B.: Positivity certificates via integral representations. Volume II of London Math Society Lecture Notes Series, pp. 84–114. Cambridge University Press, Cambridge (2022)

  17. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality. Springer (2009)

  18. Levi, F.W.: Ein Reduktionsverfahren für lineare Vektorungleichungen. Archiv der Mathematik 2, 24–26 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, M.: A determinantal inequality for positive semidefinite matrices. Electr. J. Linear Algebra 27, 821–826 (2014)

    MATH  Google Scholar 

  20. Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in \(\cal{L} _{p}\)-spaces and their applications. Studia Math. 29, 275–326 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marcus, M.: Finite Dimensional Multilinear Algebra, vol. I. Marcel Dekker (1973)

  22. Merris, R.: Multilinear Algebra. Gordon & Breach, Amsterdam (1997)

    Book  MATH  Google Scholar 

  23. Mirsky, L.: An Introduction to Linear Algebra. Oxford University Press, Oxford (1961)

    MATH  Google Scholar 

  24. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993)

    Book  MATH  Google Scholar 

  25. Nelsen, R.B.: An Introduction to Copulas. Springer (2006)

  26. Niculescu, C.P.: A new look at the Hardy-Littlewood-Pólya inequality of majorization. J. Math. Anal. Appl. 501(2), 125211 (2021)

    Article  MATH  Google Scholar 

  27. Niculescu, C.P., Olteanu, O.: From the Hahn-Banach extension theorem to the isotonicity of convex functions and the majorization theory. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas 114(4), 1–19 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

  29. Paksoy, V., Turkmen, R., Zhang, F.: Inequalities of generalized matrix functions via tensor products. Electron. J. Linear Algebra 27, 332–341 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pecaric, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Mathematics in Science and Engineering, vol. 187, Acad. Press. Inc, Boston (1992)

  31. Popoviciu, T.: Sur quelques propriétés des fonctions d’une ou de deux variables réelles. Mathematica (Cluj) 8, 1–85 (1934)

    MATH  Google Scholar 

  32. Popoviciu, T.: Sur l’approximation des fonctions convexes d’ordre supérieur. Mathematica (Cluj) 10, 49–54 (1935)

    MATH  Google Scholar 

  33. Popoviciu, T.: Les Fonctions Convexes. Hermann Cie, Paris (1944)

    MATH  Google Scholar 

  34. Popoviciu, T.: Gaz. Mat. Bucureşti. On some inequalities 51, 81–85 (1946). (Romanian)

  35. Schaefer, H.H., Wolff, M.P.: Topological Vector Spaces. Graduate Texts in Mathematics, vol. 3, Springer (1966)

  36. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions: Theory and Applications. De Gruyter, Berlin (2010)

    MATH  Google Scholar 

  37. Scott, A.D., Sokal, A.D.: Complete monotonicity for inverse powers of some combinatorially defined polynomials. Acta Math. 213, 323–392 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sendov, H.S., Zitikis, R.: The shape of the Borwein-Affleck-Girgensohn function generated by completely monotone and Bernstein functions. J. Optim. Theory Appl. 160, 67–89 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Serre, D.: The reverse Hlawka inequality in a Minkowski space. C. R. Math. Acad. Sci. 353(7), 629–633 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Siegel, C.L.: Uber die analytische Theorie der quadratischen Formen. Ann. Math. 36, 527–606 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  41. Smiley, D.M., Smiley, M.F.: The polygonal inequalities. Am. Math. Mon. 71, 755–760 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sra, S.: Reverse Minkowski (and related) determinant inequalities. MathOverflow.<RefSource /> (version: 10/10/2016)

  43. Vasić, P.M., Adamović, D.D.: Sur un système infini d’inégalités fonctionnelles. Publ. Inst. Math. Nouv. Sér. 9(23), 107–114 (1969)

    MATH  Google Scholar 

  44. Zhang, F.: Matrix Theory: Basic Results and Techniques, 2nd edn. Springer, New York (2011)

    Book  MATH  Google Scholar 

Download references

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Constantin P. Niculescu

  2. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Suvrit Sra

Authors
  1. Constantin P. Niculescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suvrit Sra
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to this work and approved the final manuscript.

Corresponding author

Correspondence to Constantin P. Niculescu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Niculescu, C.P., Sra, S. Functions with Positive Differences on Convex Cones. Results Math 78, 217 (2023). https://doi.org/10.1007/s00025-023-01987-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-023-01987-3

Keywords

Mathematics Subject Classification

Search

Navigation