1. AM is Arithmetic Mean 
The arithmetic mean is the "standard" average, often simply called the "mean". 

The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data. 

Nevertheless, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions. 

For example, the arithmetic mean of two numbers, say a and b is 

(a+b) / 2 

2. GM means Geometric Mean 

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth. 

For example, the GM of two numbers, say a and b is 

sqrt (ab) 

3. HM means Harmonic Mean 

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time). 

For example, the HM of two numbers, say a and b is 

2 / (1/a + 1/b) 

The relation between them is HM <=GM<=AM 
Proof is given in this link