5
$\begingroup$

Out of interest, I am trying to proof QM-AM-GM-HM inequality. If you don't know it, it's something like this...
Let there be $n$ numbers $x_1, x_2, x_3...x_n$, where $x_1, x_2, ...,x_n>0$.
Proof that $$\sqrt{\frac{x_1^2+x_2^2...+x_n^2}{n}}\geqslant{\frac{x_1+x_2...+x_n}{n}}\geqslant{\sqrt[n]{x_1x_2...x_n}}\geqslant{\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}...+\frac{1}{x_n}}}$$ I thought of using induction (for n). The base case was something that took me about 20 mins to solve. I used n=2 (n=1 was trivial) but I am stuck. Can anyone give me a hint to continue me? To be exact, I need help in apply the induction hypothesis to the induction step. The numbers/fractions are starting to get ... uh ... ugly... Update 1: I don't want to see the answer. Just a hint...

$\endgroup$
  • 2
    $\begingroup$ Some advice; typically the nice thing about the base case is that it should be trivial to prove, so there's nothing wrong in picking $n=1$ as your base case. Times when doing a later base case like $n=2$ are when this is used for the induction hypothesis. $\endgroup$ – Hayden Jun 20 '14 at 14:12
  • $\begingroup$ ok. Let me try... $\endgroup$ – user148697 Jun 20 '14 at 14:12
  • $\begingroup$ What does QM stand for? (Obviously it is not Quantum Mechanics; ??? mean?) $\endgroup$ – KCd Jun 20 '14 at 14:13
  • $\begingroup$ @KCd It is the Quadratic Mean. $\endgroup$ – Hayden Jun 20 '14 at 14:14
  • 3
    $\begingroup$ AM-GM inequality is a nice example of result where Cauchy induction can be used. We also have a list of proofs of AM-GM inequality on this site. $\endgroup$ – Martin Sleziak Jun 20 '14 at 14:20
1
$\begingroup$

If you're not restricted to proof by induction, you can try to show that $$ M(p; x_1,x_2,\dotsc,x_n) := \left(\frac{1}{n}\sum _{i=1} ^n x_i^p\right)^{1/p},$$ is an increasing function of $p\in\mathbb{R}$. You only need Jensen's inequality to prove this.

update: For proof without calculus, you only need to prove the AM-GM inequality (e.g., through the Cauchy induction as others suggested). QM-AM is a simple case of the Cauchy-Schwarz inequality (which has an elementary proof). Furthermore, GM-HM is the same as AM-GM for the numbers $y_i = 1/x_i$.

$\endgroup$
  • $\begingroup$ I don't really know the notation... $\endgroup$ – user148697 Jun 20 '14 at 14:36
  • $\begingroup$ @jonnytan999 What notation? I merely defined a function $M(p)$. $\endgroup$ – S.B. Jun 20 '14 at 14:37
  • $\begingroup$ QM = $M(2)$, AM = $M(1)$, GM = $\lim_{p\to 0} M(p)$, HM = $M(-1)$. $\endgroup$ – S.B. Jun 20 '14 at 14:41
  • $\begingroup$ you mean that M(p)= that, correct? $\endgroup$ – user148697 Jun 20 '14 at 14:41
  • $\begingroup$ @jonnytan999 Yes, as I mentioned particular values of $M(p)$ give you the QM, AM, GM, and HM of the $x_i$s. $\endgroup$ – S.B. Jun 20 '14 at 14:46
1
$\begingroup$

Hint for AM-GM:

Note that

$$(x_1x_2)(x_3x_4)\leq\left[\frac{x_1+x_2}{2}\right]^2\left[\frac{x_3+x_4}{2}\right]^2 \leq \left[\frac1{4}\sum_{i=1}^{4}x_i\right]^4.$$

Use this to prove by induction that

$$\left[\prod_{i=1}^{2^n}x_i\right]^{1/2^n} \leq \frac1{2^n}\sum_{i=1}^{2^n}x_i.$$

If $n$ is not a power of $2$, then choose $m$ such that $n+q=2^m$ and apply the previous result to$x_1,x_2,\ldots,x_n,A_n,\ldots,A_n$ where $A_n$ is repeated $q$ times and is the arithmetic average

$$A_n=\frac1{n}\sum_{i=1}^{n}x_i.$$

$\endgroup$
  • $\begingroup$ ok. seems convincing. What about the others? Roughly, I get the idea, but are there other methods? $\endgroup$ – user148697 Jun 20 '14 at 14:27
  • $\begingroup$ There are -- as others are showing -- but that is the straightforward induction approach. The far right inequality follows directly from AM-GM: switch $x_i$ with $1/x_i$ $\endgroup$ – RRL Jun 20 '14 at 14:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy