Characterizations of higher order strongly generalized convex functions
Consider a Hilbert space \(H\) and two transformations of the space, \(g,h:H\rightarrow H\). The authors extend the concepts of convex set and convex function by considering the behaviour of the function in the presence of the two transformations \(g\) and \(h\) of its domain. They obtain the concept of \((g,h)\)-convex subset of \(K\subseteq H\) as a set containing the point \({(1-t)h(x)}+tg(y)\), whenever \(x,y\in K\) and \(t\in [0,1]\). The properties of convexity, strong convexity, quasi-convexity, log-convexity of functions are extended in this framework of \((g,h)\)-convexity. Characterizations of each type of generalized convex functions are given. It is shown that the optimality conditions are characterized by a new class of variational inequalities in the class of generalized strongly convex functions. The operator parallelogram laws for the characterization of uniform Banach spaces are obtained as an applications of the generalized affine functions in the framework provided by the double transformation of the space using the pair \((g,h)\). For the entire collection see [Zbl 1471.34005].
scientific article; zbMATH DE number 7392150
scientific article