A New Proof of Inequalities for Two?parameter Means of Convex Function

Abstract：Suppose f (x)  is a positive differentiable function on the interval [a, b] if f′ (x)  is increasing, then for arbitrary p, q it is obtained.M p, q (f) <E (p+1?, q+1?;f (a) ?, f (b) )  () if f′ (x)  is decreasing, then the inequality () reverses.Where M p, q (f)  and E (r?, s?;a?, b)  denote the two?parameter means of function f and the extended mean values, respectively.

Keyword：power mean;generalized weighted mean values with two?parameters;two?parameter means;Tchebycheff’s integral inequality;