Inequalities, convexity and linear operators

Areas of research interest of Graham Jameson

Very many problems of classical and functional analysis are in essence questions about inequalities. The classical inequalities of Hardy, Hilbert and Copson amount to the estimation of the norms of certain very natural linear operators between Banach spaces. There are still many problems relating to generalized forms of these inequalities.

Problems of this sort sometimes equate to the determination of the supremum or infimum of the ratio beteen the tail of a series and its approximating integral, in turn leading to the problem of finding conditions under which such ratios form a monotonic sequence.

Many inequalities are derived from convexity or concavity of the functions involved. A current line of research concerns averages of the values of a convex function f at the points r/n in [ 0 , 1 ] : the average decreases with n if the end points are included, and increases with n if they are excluded. A number of inequalities previously discovered independently are special cases of these simple statements. In collaboration with G. Sinnamon (Western Ontario) and S. Abramovich (Haifa), strengthened versions of these results have been established for functions that are ``superquadratic", i.e. ``more than convex" in a certain sense.