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On Dinghas-type derivatives and convex functions of higher order. (English) Zbl 1049.26003

The authors define a function \(f: I\to \mathbb{R}\; (I\in \mathbb{R})\) to be \(T\)-convex if \[ \Delta_h^Tf(x):=\Delta_{t_1 h}\dots.\Delta_{t_n h}f(x)\geq 0 \] for \(x\in I,\; h\geq 0,\; x+(t_1 +\dots+t_n)h\in I,\) where \((t_1, \dots, t_n)=T.\) They offer among others the result that \(f\) is \(T\)-convex iff the generalized lower Dinghas derivative \[ \lim \inf_{(x,h)\to (\xi,0)\; x\leq\xi\leq x+(t_1+\dots+t_n)h}(\Delta_h ^T f(x)/(t_1\dots t_n h^n)) \] is nonnegative. The paper concludes with two open problems.

MSC:

26A51 Convexity of real functions in one variable, generalizations
26B25 Convexity of real functions of several variables, generalizations
39B62 Functional inequalities, including subadditivity, convexity, etc.
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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