Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
The Root-Mean Power-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMP-AM-GM-HM) or Exponential Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (EM-AM-GM-HM), is an inequality of the root-mean power, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers that says:
where , and is the root mean power.
The geometric mean is the theoretical existence if the root mean power equals 0, which we couldn't calculate using radicals because the 0th root of any number is undefined when the number's absolute value is greater than or equal to 1. This creates the indeterminate form of . Then, we can say that the limit as x goes to 0, the result goes to the geometric mean of the numbers.
The quadratic mean's root mean power is 2 and the arithmetic mean's root mean power is 1, as and the harmonic mean's root mean power is -1 as . Similarly, there is a root mean cube (or cubic mean), whose root mean power equals 3.
When the root mean power approaches , the mean approaches the highest number. When the root mean power reaches , the mean approaches the lowest number.
with equality if and only if . This inequality can be expanded to the power mean inequality, and is also known as the Mean Inequality Chain.
As a consequence we can have the following inequality: If are positive reals, then with equality if and only if ; which follows directly by cross multiplication from the AM-HM inequality. This is extremely useful in problem-solving.
The inequality is a direct consequence of the Cauchy-Schwarz Inequality; Alternatively, the RMS-AM can be proved using Jensen's inequality: Suppose we let (We know that is convex because and therefore ). We have: Factoring out the yields: Taking the square root to both sides (remember that both are positive):
The inequality is called the AM-GM inequality, and proofs can be found here.
The inequality is a direct consequence of AM-GM; , so , so .
Therefore the original inequality is true.
The inequality is clearly shown in this diagram for
(Note how the RMS is "sandwiched" between the minimum and the maximum)