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On the alienation of the exponential Cauchy equation and the Hosszú equation

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Abstract

In this paper, we give all the solutions \({g,h:\mathbb{R}\to\mathbb{R}}\) (the reals) of the functional equation

$$g(x)g(y)-g(x+y)=h(x+y-xy)-h(x)-h(y)+h(xy) \quad(x,y\in\mathbb{R}),$$

supposing additionally that h is continuous. This result is in connection with the alienation of the exponential Cauchy equation g(x + y) = g(x)g(y) and the Hosszú equation h(x + y−xy) + h(xy) = h(x) + h(y), namely it turns out that these equations are alien provided that h is continuous.

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References

  1. Aczél, J.: Lectures on functional equations and their applications. In: Mathematics in Science and Engineering, vol. 19. Academic Press, New York (1966)

  2. Bowers N.L. Jr., Gerber H.U., Hickman J.C., Jones D.A., Nesbitt C.J.: Actuarial Mathematics. The Society of Actuaries, Schaumburg (1997)

    MATH  Google Scholar 

  3. Daróczy Z.: On the general solution of the functional equation f(x + y−xy) + f(xy) = f(x) + f(y). Aequa. Math. 6, 130–132 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dhombres J.: Relations de dépendance entre les équations fonctionnelles de Cauchy. Aequ. Math. 35, 186–212 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fechner W.: A characterization of quadratic-multiplicative functionals. Monatsh. Math. 164, 383–392 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fechner W.: A note on alienation for functional inequalities. J. Math. Anal. Appl. 385, 202–207 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ger R.: On an equation of ring homomorphisms. Publ. Math. Debr. 52(3-4), 397–412 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Ger R.: Ring homomorphisms equation revisited. Rocz. Nauk.-Dydakt. Akad. Pedag. Krak. Prace Mat. 17, 101–115 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Ger R.: Additivity and exponentiality are alien to each other. Aequa. Math. 80, 111–118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ger R.: The alienation phenomenon and associative rational operations. Ann. Math. Sil. 27, 75–88 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Ger R.: Alienation of additive and logarithmic equations. Ann. Univ. Sci. Bp. Sect. Comp. 40, 269–274 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Ger R., Reich L.: A generalized ring homomorphisms equation. Monatsh. Math. 159, 225–233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gerber H.U.: An Introduction to Mathematical Risk Theory. Homewood, Philadelphia (1979)

    MATH  Google Scholar 

  14. Gselmann E.: Notes on the characterization of derivations. Acta Sci. Math. (Szeged) 78(1–2), 137–145 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Heilpern S.: A rank-dependent generalization of zero utility principle. Insur. Math. Econ. 33, 67–73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kałuszka M., Krzeszowiec M.: Pricing insurance contracts under cumulative prospect theory. Insur. Math. Econ. 50, 159–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kuczma, M.: An introduction to the theory of functional equations and inequalities. In: Gilányi, A. (ed.) Prace Naukowe Uniwersytetu Śla̧skiego w Katowicach, Państwowe Wydawnictwo Naukowe Uniwersytet Śla̧ski, Warszawa-Kraków-Katowice, 1985, vol. 489. 2nd edn. Birkhäuser, Basel (2009)

  18. Sablik, M.: Alienating functional equations. In: Talk at the 51st ISFE, Rzeszów, Poland 2013 in Aequationes Math., Report on the 51st International Symposium on Functional Equations, vol. 80, no. 3, pp. 291–320 (2014)

  19. Sablik, M.: Additivity of insurance premium. In: Talk at Positivity VII, Zaanen Centennial Conference, Leiden, Netherlands, July 22–26 (2013)

  20. Sablik, M.: On alien functional equations. Talk at the Special Session, 52nd ISFE, Innsbruck, Austria 2014. In: Aequationes Math., Report on the 52nd International Symposium on Functional Equations (2015) (to be published)

  21. Tyrala I.: Solutions of the Dhombres-type trigonometric functional equation. Sci. Issues Jan D lugosz Univ. Czest. Math. 16, 87–94 (2011)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, Pf. 12, Debrecen, 4010, Hungary

    Gyula Maksa

  2. Institute of Mathematics, University of Silesia in Katowice, Bankowa Street 14, 40-007, Katowice, Poland

    Maciej Sablik

Authors
  1. Gyula Maksa
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  2. Maciej Sablik
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Correspondence to Gyula Maksa.

Additional information

Dedicated to Professor Roman Ger on the occasion of his 70th birthday

The research of the first author was supported by the Hungarian Scientific Research Fund (OTKA) Grant NK 111651.

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Maksa, G., Sablik, M. On the alienation of the exponential Cauchy equation and the Hosszú equation. Aequat. Math. 90, 57–66 (2016). https://doi.org/10.1007/s00010-015-0358-y

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  • DOI: https://doi.org/10.1007/s00010-015-0358-y

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