The parts of these statements about mean and harmonic mean are not true for sets containing negative numbers.

Consider the set (-1, -2, -3, -4, -5). The mean is -3, and the harmonic mean is -2.19.

For the set (-2, -1, 0, 1, 2), both the mean and the harmonic mean are 0.

Also, the trivial cases where all the numbers are the same, e.g. (2, 2, 2, 2, 2)

GM=√(AM×HM)

So, AM=GM^2/HM means AM is greater than GM

HM=GM^2/AM means HM tends to be smaller than GM

thank you very much prof. Sourav Maiti

The parts of these statements about mean and harmonic mean are not true for sets containing negative numbers.

Consider the set (-1, -2, -3, -4, -5). The mean is -3, and the harmonic mean is -2.19.

For the set (-2, -1, 0, 1, 2), both the mean and the harmonic mean are 0.

Also, the trivial cases where all the numbers are the same, e.g. (2, 2, 2, 2, 2)

For positive data, A>=G>=H. If the data are 1, 4, 7 then the Arithmetic mean=4, Geometric mean = 3.0366, Harmonic mean = 2.1538. If the data are 2, 2, 2 then the means are equal. They are also equal if the data are -2, -2, -2.

If the data are 1, -4, 7 then the arithmetic mean=1.33, geometric mean=-3.037, and harmonic mean= 3.36.

Try the wikipedia pages. Here is the one for harmonic mean: https://en.wikipedia.org/wiki/Harmonic_mean

Here is the one for geometric mean: https://en.wikipedia.org/wiki/Geometric_mean

It has been established mathematically that

Arithmetic Mean > (& =) Geometric Mean > (& =) Harmonic Mean

Arithmetic Mean , Geometric Mean and Harmonic Mean satisfy the following relationship:

Arithmetic Mean > (& =) Geometric Mean > (& =) Harmonic Mean

Mean, median and mode satisfy the theoretical relationship

Mean = Median = Mode

if the associated distribution is symmetric.

For a moderately asymmetric distribution mean , median and mode follows the empirical relationship (due to Kearl Pearson)

3 (mean - median) = mean - mode

Note that this relationship is empirical but not theoretical.

In the case of Arithmetic Mean , Geometric Mean and Harmonic Mean , it has not yet been possible to establish such theoretical relationship and/or such empirical relationship.

Dhritikesh Chakrabarty , the mean can be less than the harmonic mean, for example for (-1, -2, -3, -4, -5), the mean is -3, and the harmonic mean is -2.19.

This example was given in my earlier answer.

Dear Salvatore S. Mangiafico,

Your tricky counter example is correct.

The inequality

Arithmetic Mean > (& =) Geometric Mean > (& =) Harmonic Mean

is true if and only if all the three means exist.

For the existence of these three means, the observations are required to be non-zero and non negative.

Thus the above inequality holds good for non-zero and non negative observations.

In your counter example, the observations are outside of this domain.

I do recommend your tricky example.

Salvatore S. Mangiafico has given a counter example of the inequality

" Arithmetic Mean > (& =) Geometric Mean > (& =) Harmonic Mean".

I do recommend his answer.

Srikanth Potharla, have a look at...

https://en.wikipedia.org/wiki/Generalized_mean

Cheers!

It is not true that

(1) Arithmetic Mean is always more than geometric mean and harmonic mean

& (2) Harmonic mean is always lower than arithmetic mean and geometric mean.

But, it is true that

(1) Arithmetic Mean is more than geometric mean and harmonic mean

& (2) Harmonic mean is lower than arithmetic mean and geometric mean

only if the values (or the numbers or the observations) whose means are to calculated are real and strictly positive.