QM–AM–GM–HM inequalities

For positive numbers, the quadratic mean is at least the arithmetic mean, which is at least the geometric mean, which is at least the harmonic mean. Equality throughout holds when all numbers are equal.

Connect

Cauchy–Schwarz inequality: An inequality bounding the inner product of two vectors by the product of their lengths. Applied to the numbers and to their reciprocals, it establishes the quadratic-mean and harmonic-mean bounds.
Quadratic mean: The square root of the mean of the squares of a collection of numbers. It is the largest of the four means in the stated chain.
Root mean square: A measure of magnitude equal to the square root of the average of squared values. It is the quadratic mean used in alternating-current measurements and signal analysis.
Weighted arithmetic mean: An average formed by assigning nonnegative weights to values and dividing by the total weight. The stated chain uses equally weighted inputs; weighted versions require corresponding weighted means.
Jensen's inequality: An inequality relating a convex function of an average to the average of that function. Convexity of the exponential function yields the arithmetic–geometric mean bound.
Arithmetic mean: The sum of a collection of numbers divided by their count. It sits between the quadratic and geometric means.
Harmonic mean in physics: Use of the harmonic mean to summarize quantities such as speeds over equal distances. The chain helps explain why averaging rates requires care about how they are sampled.
Weighted geometric mean: A product of positive values raised to nonnegative weights that sum to one. It extends the geometric mean beyond equal contributions, unlike the unweighted chain.
Power mean inequality: A theorem stating that generalized means increase as their exponent increases. It places all four means in one ordered family and gives the full chain directly.
Geometric mean: The nth root of the product of n positive numbers. It occupies the middle of the chain and connects additive and multiplicative averaging.