Convex sets and inequalities (Q2576033)

Let \(X\) be an arbitrary set and \(\phi_o,\phi_1,\phi\) be a triple of nonnegative real-valued functions on \(X\). Let \[ m= \inf_{\phi_0(x)\neq 0} \frac{\phi(x)}{\phi_0(x)}, \] \[ M= \sup_{\phi_1(x)\neq 0} \frac{\phi(x)}{\phi_1(x)}. \] Suppose that \(0<m,M<\infty\). For each \(x\in X\), let \[ D_{\phi}(x)=\{(\alpha, \beta)\in R^2: \phi(x) \leq \alpha\phi_1(x)+\beta\phi_0(x)\}. \] The authors consider the intersection \(D_{\phi}=\bigcap_{x\in X}D_{\phi}(x)\); \(D_{\phi}\) is a nonempty closed convex domain in the plane. They study some basic properties of \(D_{\phi}\). Using these properties for particular choices of the set \(X\) and of the functions \(\phi_o,\phi_1,\phi\) they obtain new types of power mean inequalities and of Hölder-type inequalities.