In this article, all of the cases of the inequality b(x+y) - a(x+y)/b(
x) - a(x) greater than or equal to x + y/x (a + b/2)(y) being or not b
eing valid are discussed for all real numbers x,y is an element of R a
nd x not equal 0, where a, b > 0 and a not equal b. Some recent develo
pments concerning mean values are simply introduced. (C) 1998 Academic
Press.