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Equality of two-variable functional means generated by different measures

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Abstract

We consider two-variable functional means of the form

$$M_{f,g;\mu}(x,y) := \left(\frac{f}{g}\right)^{-1}\left(\frac{\int\nolimits_{[0,1]} f(tx+(1-t)y)\,d\mu(t)}{\int\nolimits_{[0,1]}g(tx+(1-t)y)\,d\mu(t)}\right),$$

where f, g are continuous functions on a real interval such that g is positive, f/g is strictly monotonic and μ is a measure over the Borel sets of [0,1]. The main results concern the functional equation M f,g;μ = M f,g;ν for the unknown functions f, g, where μ and ν are given measures. Depending on the symmetry properties of the measures, various necessary conditions and sufficient conditions are established.

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References

  1. Aczél J., Daróczy Z.: Über verallgemeinerte quasilineare Mittelwerte, die mit Gewichtsfunktionen gebildet sind. Publ. Math. Debrecen 10, 171–190 (1963)

    MathSciNet  Google Scholar 

  2. Alzer H.: Bestmögliche Abschätzungen für spezielle Mittelwerte. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23(1), 331–346 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Bajraktarević M.: Sur une équation fonctionnelle aux valeurs moyennes. Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 13, 243–248 (1958)

    Google Scholar 

  4. Bajraktarević M.: Über die Vergleichbarkeit der mit Gewichtsfunktionen gebildeten Mittelwerte. Studia Sci. Math. Hungar. 4, 3–8 (1969)

    MathSciNet  MATH  Google Scholar 

  5. Berrone L.R.: The mean value theorem: functional equations and Lagrangian means. Epsilon 14((1(40))), 131–151 (1998)

    MathSciNet  Google Scholar 

  6. Berrone L.R., Moro J.: Lagrangian means. Aequationes Math. 55(3), 217–226 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenner J.: A unified treatment and extension of some means of classical analysis. I. Comparison theorems. J. Combin. Inform. System Sci. 3, 175–199 (1978)

    MathSciNet  MATH  Google Scholar 

  8. Carlson B.C.: The logarithmic mean. Amer. Math. Monthly 79, 615–618 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Járai, A.: Regularity Properties of Functional Equations in Several Variables, Adv Math, vol. 8. Springer, Berlin (2005)

  10. Losonczi L.: Inequalities for integral mean values. J. Math. Anal. Appl. 61(3), 586–606 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Losonczi L.: Equality of two variable weighted means: reduction to differential equations. Aequationes Math. 58(3), 223–241 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Losonczi L.: Equality of Cauchy mean values. Publ. Math. Debrecen 57, 217–230 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Losonczi L.: Equality of two variable Cauchy mean values. Aequationes Math. 65(1–2), 61–81 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Losonczi L.: Equality of two variable means revisited. Aequationes Math. 71(3), 228–245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Losonczi L., Páles Zs.: Minkowski’s inequality for two variable difference means. Proc. Amer. Math. Soc. 126(3), 779–789 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Losonczi L., Páles Zs.: Comparison of means generated by two functions and a measure. J. Math. Anal. Appl. 345(1), 135–146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leach E., Sholander M.: Extended mean values. Amer. Math. Monthly 85(2), 84–90 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leach E., Sholander M.: Extended mean values. II. J. Math. Anal. Appl. 92, 207–223 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leach E., Sholander M.: Multivariable extended mean values. J. Math. Anal. Appl. 104, 390–407 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Makó Z., Páles Zs.: On the equality of generalized quasiarithmetic means. Publ. Math. Debrecen 72, 407–440 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series), vol. 53. Kluwer Academic Publishers Group, Dordrecht (1991)

  22. Mitrinović D.S., Pečarić J.E., Fink A.M.: Classical and New Inequalities in Analysis, Mathematics and its Applications (East European Series), vol. 61. Kluwer Academic Publishers Group, Dordrecht (1993)

    Google Scholar 

  23. Páles Zs.: On the characterization of quasi-arithmetic means with weight function. Aequationes Math. 32(2–3), 171–194 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Páles Zs.: Comparison of two variable homogeneous means, General Inequalities 6 (Oberwolfach, 1990). In: Walter, W. (eds) International Series of Numerical Mathematics, pp. 59–70. Birkhäuser, Basel (1992)

    Google Scholar 

  25. Sándor J.: On certain inequalities for means. J. Math. Anal. Appl. 189, 602–606 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stolarsky K.B.: Generalizations of the logarithmic mean. Math. Mag. 48, 87–92 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stolarsky K.B.: The power and generalized logarithmic means. Amer. Math. Monthly 87(7), 545–548 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Economics and Business Administration, Debrecen University, Kassai u. 26, 4028, Debrecen, Hungary

    László Losonczi

  2. Institute of Mathematics, Debrecen University, PF. 12, 4010, Debrecen, Hungary

    Zsolt Páles

Authors
  1. László Losonczi
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  2. Zsolt Páles
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Correspondence to László Losonczi.

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Dedicated to the 85th birthday of Professor János Aczél

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK81402 and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project implemented through the New Hungary Development Plan co-financed by the European Social Fund, and the European Regional Development Fund.

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Losonczi, L., Páles, Z. Equality of two-variable functional means generated by different measures. Aequat. Math. 81, 31–53 (2011). https://doi.org/10.1007/s00010-010-0059-5

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  • DOI: https://doi.org/10.1007/s00010-010-0059-5

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