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On convex triangle functions

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Abstract

We prove that the strongest (largest convex) solution of the functional inequality

$$\tau \left( {\frac{{F + G}}{2},\frac{{H + K}}{2}} \right) \le \frac{{\tau (F,H) + \tau (G,K)}}{2},$$

whereF, G, H andK are arbitrary distribution functions, is the triangle function τ(F, G)(x) = Max(F(x) +G(x) − 1, 0).

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References

  1. Alsina, C.,On countable products and algebraic convexifications of probabilistic metric spaces. Pacific J. Math.76 (1978), 291–300.

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  2. Alsina, C.,On a family of functional inequalities. InGeneral Inequalities 2, Birkhäuser, Basel 1981, pp. 419–427.

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  3. Schweizer, B. andSklar, A.,Probabilistic metric spaces. Elsevier, North Holland-New York, 1983.

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Authors and Affiliations

  1. Department de Matemàtiques i Estadística, E.T.S. Arquitectura de Barcelona, Universitat Politècnica de Barcelona, Diagonal 649, Barcelona -28-, Spain

    Claudi Alsina

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  1. Claudi Alsina
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Alsina, C. On convex triangle functions. Aeq. Math. 26, 191–196 (1983). https://doi.org/10.1007/BF02189682

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  • DOI: https://doi.org/10.1007/BF02189682

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