Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMS-AM-GM-HM), is an inequality of the root-mean square, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers that says:
with equality if and only if . This inequality can be expanded to the power mean inequality.
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.
Proof
The inequality is a direct consequence of the Cauchy-Schwarz Inequality;
, so
, so
.
Alternatively, the RMS-AM can be proved using Jensen's inequality:
Suppose we let (We know that
is convex because
and therefore
).
We have:
$F\left(\frac{x_1}{n}+\cdots+\frac{x_n}{n}\right)\le \frac1)}{n}+\cdots+\frac{F(x_n)}{n}$ (Error compiling LaTeX. Unknown error_msg);
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 Root Mean Square is also known as the quadratic mean, and the inequality is therefore sometimes known as the QM-AM-GM-HM Inequality.
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