On a separation for the Cauchy equation on spheres

We give sufficient conditions for separating functions p and q, satisfying for a given function gamma the conditional inequalities g (x) = gamma (y) double right arrow p(x + y) <= p(x) + p(y), g (x) = gamma (y) double right arrow q(x + y) >= q(x) + q(y), by an additive mapping. (c) 2012 Elsevier Ltd. All rights reserved.