On Some Properties of Functions on Convex Galaxies

Abstract

In this paper, we define and study extensively a new type of external sets in , we call it "convex galaxies". We show that these convex external sets may be classified in some definite types. More precisely, we obtain the following : (1)Let  be a convex galaxy which is symmetric with respect to zero, then (i)  is an  - galaxy (0) if and only if there exists an internal strictly increasing sequence of strictly positive real numbers  with  such that  and , for all , where,  is some limited real number such that . (ii)  is a non-linear galaxy if and only if there exists an internal strictly increasing sequence of strictly positive real numbers  with   such that  is unlimited for all .  (2)Let  be a convex galaxy which is symmetric with respect to zero, then (i)  is an  - galaxy (0) iff there exists a real internal strictly increasing - function , such that , and  for all limited , where  is a positive real number. (ii)  is a non-linear galaxy if and only if there exists a real internal strictly increasing - function , such that  and  is positive unlimited, for all appreciable .