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Results on \(t\)-Wright convexity. (English) Zbl 0749.26007
J. Matkowski asked about the relations between \(t\)-Wright convexity and Jensen convexity. A function \(f\) from a convex set \(D\) in a real vector space into \(\mathbb{R}\) is said \(t\)-Wright convex for a fixed \(t\in(0,1)\) if \[ f(tx+(1-t)y)+f((1-t)x+ty)\leq f(x)+f(y),\quad x,y\in D. \] In the paper under review it is proved that if \(f\) is \(t\)-Wright convex for a rational \(t\), then it is Jensen convex.
Conversely, if \(t\) is transcendental or has one of its algebraic conjugates outside the open disk \(\{z\in\mathbb{C}: | z-1/2|<1/2\}\), then there exists a \(t\)-Wright convex function \(f:\mathbb{R}\to(-\infty,0]\) which is not Jensen convex.
Reviewer: G.L.Forti (Milano)

26B25 Convexity of real functions of several variables, generalizations