Hermite-Hadamard inequalities for generalized convex functions
Let \(\mathbf{\omega}=(\omega_{1},\ldots,\omega_{n})\) be a Tchebychev system on an interval \([a,b].\) A function \(f:[a,b]\rightarrow\mathbb{R}\) is called generalized \(n\)-convex with respect to \(\mathbf{\omega}\) if for all \(x_{0}<\cdots<x_{n}\) in \(I\) we have \[ (-1)^{n}\left| \begin{matrix} f(x_{0}) & \ldots & f(x_{n})\\ \omega_{1}(x_{0}) & \ldots & \omega_{1}(x_{n})\\ \ldots& \ldots &\ldots \\ \omega_{n}(x_{0}) &\ldots& \omega_{n}(x_{n}) \end{matrix} \right| \geq0 \] [see \textit{S. Karlin} and \textit{W. J. Studden}, ``Tchebychev system: With applications in analysis and statistics'' (1966; Zbl 0153.38902)]. The main result of the paper under review asserts that \(\mathbf{c}=\int _{a}^{b}\rho\mathbf{\omega}\) is an interior point of the moment space of \(\mathbf{\omega},\) whenever \(\rho\) is a positive integrable function on \([a,b].\) This is used to establish Hermite-Hadamard type inequalities for generalized \(n\)-convex functions [see also \textit{M. Bessenyei} and \textit{Zs. Páles}, Acta Sci. Math. 70, No. 1--2, 13--24 (2004; Zbl 1069.26010)].
scientific article; zbMATH DE number 2162491