A note on mean inequalities
Introduction
Throughout this note, we consider mean of
Next, we introduce the so-called power mean. We define a family of functions
The following facts are well known:
-
[F1]
-
[F2]
-
[F3] As a function of
, is increasing when increases.
The above fact [F3] implies that
In this note, based on the above, an inequality is derived in Theorem 1, which, I believe, deserves to be widely known. Two examples are shown as special cases of Theorem 1.
Main results
Theorem 1:
If
Proof:
Take an arbitrary positive integer
thus taking limit on the both sides leads to the desired result.
Example 1: Using Theorem 1 and noticing
Example 2: Since
References
[1] Miguel de Carvalho, Mean, What do You Mean? The American Statistician 70 (2016) 270–274.