Sharp bounds for the difference between the arithmetic and geometric means

J. M. Aldaz

We present sharp bounds for ${\sum }_{i=1}^n{\alpha }_i{x}_i-{\prod }_{i=1}^n{x}_i^{{\alpha }_i}$ in terms of the variance of the vector $(x_1^{1/2},...,x_n^{1/2})$.