Abstract and Applied Analysis
Volume 2008 (2008), Article ID 136592, 11 pages
Research Article

Functional Inequalities Associated with Additive Mappings

Jaiok Roh1 and Ick-Soon Chang2

1Department of Mathematics, Hallym University, Chuncheon 200-702, South Korea
2Department of Mathematics, Mokwon University, Daejeon 302-729, South Korea

Received 13 May 2008; Revised 25 June 2008; Accepted 1 August 2008

Academic Editor: John Rassias

Copyright © 2008 Jaiok Roh and Ick-Soon Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The functional inequality f(x)+2f(y)+2f(z)2f(x/2+y+z)+ϕ  (x,y,z)(x,y,zG) is investigated, where G is a group divisible by 2,f:GX and ϕ:G3[0,) are mappings, and X is a Banach space. The main result of the paper states that the assumptions above together with (1) ϕ(2x,x,0)=0=ϕ(0,x,x) (xG) and (2) limn(1/2n)ϕ(2n+1x,2ny,2nz)=0, or limn2nϕ(x/2n1,y/2n,z/2n)=0  (x,y,zG), imply that f is additive. In addition, some stability theorems are proved.