Ф-variation and p-variation; Inequalities for Integrals

Abstract

Let V be the class of all functions Φ: [0,∞) ? [0,∞) which are strictly increasing, continuous, unbounded, and 0 at 0. Let CV be the subclass of convex functions in V. Let X be a Banach space with norm $\|\cdot\|$, let J be a nonempty interval in R, let f be a function defined on J with values in X, and let Φ ∈ V. Recall that an interval is called nondegenerate if it contains more than one point.