A Generalization of Two Inequalities Involving Means (original) (raw)

Fan has proven an inequality relating the arithmetic and geometric means of (x¡, • • • , x") and (1-*,, • • •, 1-x"), where 0<x,^|, i=l, ■ ■ ■ ,n. Levinson has generalized Fan's inequality; his result involves functions with positive third derivatives on (0, 1). In this paper, the above condition that requires 0<JC¡ísi has been replaced by a condition which only weights the x¡ to the left side of (0, 1) in pairs, and Levinson's differentiability requirement has been replaced by the analogous condition on third differences.