A generalization of two inequalities involving means

Abstract:

Fan has proven an inequality relating the arithmetic and geometric means of $({x_1}, \cdots ,{x_n})$ and $(1 - {x_1}, \cdots ,1 - {x_n})$, where $0 < {x_i} \leqq \tfrac {1}{2},i = 1, \cdots ,n$. 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 $0 < {x_i} \leqq \tfrac {1}{2}$ has been replaced by a condition which only weights the ${x_i}$ to the left side of (0, 1) in pairs, and Levinson’s differentiability requirement has been replaced by the analogous condition on third differences.