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On convex triangle functions. (English) Zbl 0564.39006

The author considers the class \(\Delta^+\) of probability distribution functions on the real line, which are left continuous and vanish at the origin. If \(\epsilon_ a\in \Delta^+\) has the single point a of discontinuity, \(a\geq 0\), and if the function \(\tau: \Delta^+\times \Delta^+\to \Delta^+\) has the properties \((i)\quad \tau (F,\epsilon_ 0)=F,\) (ii) \(\tau\) (F,G)\(\leq \tau (H,K)\), if \(F\leq H\), \(G\leq K\), (iii) \(\tau (F,G)=\tau (G,F)\), \((iv)\quad \tau (\tau (F,G),H)=\tau (F,\tau (G,H)),\) \(F,G,H,K\in \Delta^+\), then the largest solution of the inequality \(\tau ((F+G)/2,(H+K)/2)\leq (\tau (F,H)+\tau (G,K))/2\) is \(\max (F+G-1,0)\).
Reviewer: A.B.Buche

MSC:

39B72 Systems of functional equations and inequalities
60E05 Probability distributions: general theory
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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. · Zbl 0412.54007
[2] Alsina, C.,On a family of functional inequalities. InGeneral Inequalities 2, Birkhäuser, Basel 1981, pp. 419–427.
[3] Schweizer, B. andSklar, A.,Probabilistic metric spaces. Elsevier, North Holland-New York, 1983.
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