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ascky
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Does anyone know where I can find (online?) the proofs for the arithmetic mean-geometric mean and the power mean inequalities? Thanks in advance.
Gokul43201 said:If you want a proof of AM > GM, it's pretty simple :
Start with :
[tex](a-b)^2 > 0, ~ a <> b [/tex]
and work your way towards
[tex]=>a + b > 2 \sqrt{ab} [/tex]
What is the power mean inequality - can you write it down ?
ascky said:Grr not sure how to use Latex. I'll give it a stab.
I get why [tex]AM \geq GM[/tex] for two variables, but what I don't get is how to prove it still works when you extend it to n variables.
Power mean inequality:
[tex]P_a=((x^a_1...x^a_n)/n)^{1/a}[/tex], where [tex]x_1,...,x_n \geq 0[/tex]
Then if [tex]a>b[/tex], [tex]P_a \geq P_b[/tex]
I think the power mean inequality thing should be obvious... hmm. I mean, I guess you could just subsitute values right?
Erp. You're absolutely right... forget that.maverick280857 said:Are you sure you want to prove that Pa >Pb?
The AM-GM inequality, also known as the arithmetic mean-geometric mean inequality, states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean of the same set of numbers.
The power mean inequality is a more general form of the AM-GM inequality that states for any real number r, the rth power mean of a set of non-negative numbers is always greater than or equal to the arithmetic mean of the same set of numbers. The special case of r=1 corresponds to the AM-GM inequality.
The proof for the AM-GM and power mean inequality involves using mathematical induction and the Cauchy-Schwarz inequality. It can be found in most textbooks on inequalities or can be easily found online.
The AM-GM inequality is a useful tool in mathematical analysis, especially in problems involving optimization and inequalities. It is also used in various fields such as physics and economics.
The AM-GM inequality is important because it is a fundamental inequality in mathematics that has many applications and can be used to prove other important inequalities. It also helps in understanding the relationship between the arithmetic and geometric means of a set of numbers.