proof of inequalities for difference of powers

1 First Inequality

We have the factorization

$u^{n}-v^{n}=(u-v)\sum_{k=0}^{n-1}u^{k}v^{n-k-1}.$

Since the largest term in the sum is is $u^{n-1}$ and the smallest is $v^{n-1}$ , and there are $n$ terms in the sum, we deduce the following inequalities:

$n(u-v)v^{n-1}<u^{n}-v^{n}<n(u-v)u^{n-1}$

2 Second Inequality

This inequality is trivial when $x=0$ . We split the rest of the proof into two cases.

2.1 $-1<x<0$

In this case, we set $u=1$ and $v=1+x$ in the second inequality above:

$1-(1+x)^{n}<n(-x)$

Reversing the signs of both sides yields

$nx<(1+x)^{n}-1$

2.2 $0<x$

In this case, we set $u=1+x$ and $v=1$ in the first inequality above:

$nx<(1+x)^{n}-1$

3 Third Inequality

This inequality is trivial when $x=0$ . We split the rest of the proof into two cases.

3.1 $-1<x<0$

Start with the first inequality for differences of powers, expand the left-hand side,

$nuv^{n-1}-nv^{n}<u^{n}-v^{n},$

move the $v^{n}$ to the other side of the inequality,

$nuv^{n-1}-(n-1)v^{n}<u^{n},$

and divide by $v^{n}$ to obtain

$n{u\over v}-n+1<\left({u\over v}\right)^{n}.$

Taking the reciprocal, we obtain

$\left({v\over u}\right)^{n}<{v\over v+n(u-v)}=1-{n(u-v)\over v+n(u-v)}$

Setting $u=1$ and $v=1+x$ , and moving a term from one side to the other, this becomes

$(1+x)^{n}-1<{nx\over 1-(n-1)x}.$

3.2 $0<x<1/(n-1)$

Start with the second inequality for differences of powers, expand the right-hand side,

$u^{n}-v^{n}<nu^{n}-nu^{n-1}v$

move terms from one side of the inequality to the other,

$nu^{n-1}v-(n-1)u^{n}<v^{n}$

and divide by $u^{n}$ to obtain

$n{v\over u}-n+1<\left({v\over u}\right)^{n}$

When the left-hand side is positive, (i.e. $nv>(n-1)u$ ) we can take the reciprocal:

$\left({u\over v}\right)^{n}<{u\over u-n(u-v)}=1+{n(u-v)\over u-n(u-v)}$

Setting $u=1+x$ and $v=1$ , and moving a term from one side to the other, this becomes

$(1+x)^{n}-1<{nx\over 1-(n-1)x}$

and the positivity condition mentioned above becomes $(n-1)x<1$ .

Title	proof of inequalities for difference of powers
Canonical name	ProofOfInequalitiesForDifferenceOfPowers
Date of creation	2013-03-22 15:48:45
Last modified on	2013-03-22 15:48:45
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	14
Author	Mathprof (13753)
Entry type	Proof
Classification	msc 26D99