One-sided estimates for quasimonotone increasing functions

Let E be a Banach space ordered by a solid and normal cone K, and normed by the Minkowski functional of an order interval [-p, p], p is an element of Kdegrees. We derive global one-sided estimates for quasimonotone increasing functions f : [0, T) x E --> E with respect to the norm, and the distance to the line generated by p, under conditions on f in direction p.