On convex triangle functions (Q1058078)

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 \(\max (F+G-1,0)\).