LOWER AND UPPER BOUNDS FOR POSITIVE LINEAR FUNCTIONALS

This paper deals with the problem of finding lower and upper-bounds in a set of convex functions to a given positive linear functional; that is, bounds which estimate always below (or above) a functional over a family of convex functions. A new set of upper and lower bounds are provided and their extremal properties are established. Moreover, we show how such bounds can be combined to produce better error estimates. In addition, we also extend many results from [7], which hold true for simplices, to results for any convex polytopes. Particularly, we use our result to obtain multivariate versions of some inequalities first given, respectively, by Favard in [3] and Hammer in [14], over any convex polytope. For smooth (nonconvex) twice continuously differentiable functions, we will also show how both the lower and upper bounds could be improved. Finally, we establish a general result concerning error estimates. This seems to suggest a more unified and effective approach for problems of this sort.