Several problems about the generalized convexity (Q2732209)

Let \(r>0\). The author says that a function \(f:\mathbb{R}^n \to\mathbb{R}\) is \(r\)-convex (respectively: analog convex) if for every \(x_1,x_2 \in\mathbb{R}^n\) and for every \(q_1\geq 0\), \(q_2\geq 0\) such that \(q_1+q_2=1\) we have \(f(q_1x_1+ q_2 x_2)\leq \ln[q_1e^{r\cdot f(x_1)}+ q_2e^{r\cdot f(x_2)}]^{1/r}\) (respectively: \(f (q_1x_1+ q_2x_2)\leq \max\{f(x_1), f(x_2)\})\). He shows that if \(r<s\), then any \(r\)-convex function is \(s\)-convex. He also concludes that every \(r\)-convex function is analog convex.NEWLINENEWLINENEWLINEReviewer's remark: The definition of \(r\)-convex function may be given by the simpler condition that the function \(e^{r \cdot f(x)}\) is convex.