关于凸函数的幂平均不等式的一点注记

NOTE ON THE MEAN VALUE INEQUALITY OF THE POWER ABOUT THE CONVEX FUNCTION

杨镇杭 [1]曾得到如下结论 :f(x) >0 ,x ∈ [a ,b],且 f′′(x)存在 ,则 (1)当 f′′(x) >0 ,α≥ 1时有f(a +b2 ) <Mx(f) <Xα(f(a) ,f(b) ) ;(2 )当 f′′(x) <0时 ,不等式反向。本文指出结论 (2 )是不正确的 ,并且得到了当α≤ 1时凹函数的幂平均不等式。同时讨论了当α取其它值时不等式的情况。

Yang Zhenhang [1] has reached this conclusion:f(x)>0,(1)f(a+b2)<M α(f)<x α(f(a),f(b)) when f′′(x)>0,α≤1;(2)This inequality is opposite when f′′(x)<0.This paper points out the incorrectness of the conclusion(2),and has obtained the mean value inequality of power about the concave function when α≤1,and also discusses the mean value inequality of power in other cases of α.