几何凸函数的对称拟算术平均不等式

The Mean Inequality for Quasi-arithmetic Symmetrical Mean of Geometrically Convex Functions

建立了几何凸函数的对称拟算术平均不等式,对文献[1]提出的不等式进行了推广统一;引进加权对数幂平均的概念,建立起其与双参数平均之间的关系,得到加权对数平均不等式,从而确定了几何凸函数的几何平均、算术平均的上界的大小关系;最后,提出了几何凸函数的对称拟算术平均不等式的推广问题。

An inequality for quasi-arithmetic symmetrical mean of geometrically convex functions is established, and inequalities presented by article [ 1 ] are unified and generalized. The concept of the weighted logarithmic power mean is introduced; its relation with two-parameter mean is given; the inequality for weighted logarithmic power mean is derived; the magnitude relation among upper bounds of geometric mean and arithmetic mean of geometrically convex functions are made certain. A problem of generalization on inequality for quasi-arithmetic symmetrical mean of geometrically convex functions is put forward.