IIT JEE , AIEEE Forum - Mathematics , Algebra - The Inequalities thread (Theorems/Axioms)

Conjurer

OK I want to start a thread where everyone can post all the inequalities which havent been posted already.So this can be a nice reference thread.Please contribute.So I begin with the most basic:

1) QM

AM

GM

HM ,equality holding iff x₁ = x₂= x₃= ... = x_n
i.e

root[(x_1²+ x_2²+ .......+x²_n)/n] >= (x₁+ x₂+ ..... + x_n) /n >= ^{[ n]}

x₁x₂x_3....x_n>= [(1/x₁+ 1/x₁ +1/x₃+......1/x_n)/n]^-1

konichiwa2x

nice idea!

intitial content moved to http://www.goiit.com/posts/list/algebra-the-inequality-question-thread-46974.htm

Conjurer

Erm I would suggest not posting questions here as new questions wont be visible easily.Let this just be a thread for reference.You may post examples with your inequality though.

hsbhatt

Look at these posts. http://www.goiit.com/posts/list/trignometry-prove-the-following-inequality-29628.htm#212552

vineetnegi

I wonder that the rearrange ment inequality and all other inequality can be prooved by induction
i tried for rearr. inequality
but want to know about
AM GM one
can someone help??????????/

konichiwa2x

Chebyshev's inequality

Given reals $a_1 \ge a_2 \ge \ldots \ge a_n \ge 0$ and $b_1 \ge b_2 \ge \ldots \ge b_n$ , we have

${{\sum a_ib_i}\over{n}} \ge {{\sum a_i}\over{n}}{{\sum b_i}\over{n}}$ .

Minkowsky's inequality

Given reals $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ , we have

$\sqrt{\sum a_i^2} + \sqrt{\sum b_i^2} \ge \sqrt{\sum (a_i+b_i)^2}$

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