Contraharmonic mean

In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L p {displaystyle L_{p}} , where p = 2. In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L p {displaystyle L_{p}} , where p = 2. The contraharmonic mean of a set of positive numbers is defined as the arithmetic mean of the squares of the numbers divided by the arithmetic mean of the numbers: It is easy to show that this satisfies the characteristic properties of a mean: The first property implies the fixed point property, that for all k > 0, The contraharmonic mean is higher in value than the arithmetic mean and also higher than the root mean square: where x is a list of values, H is the harmonic mean, G is geometric mean, L is the logarithmic mean, A is the arithmetic mean, R is the root mean square and C is the contraharmonic mean. Unless all values of x are the same, the ≤ signs above can be replaced by <. The name contraharmonic may be due to the fact that when taking the mean of only two variables, the contraharmonic mean is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean (i.e., the arithmetic mean of the two variables is equal to the arithmetic mean of their harmonic and contraharmonic means).