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:

2 Second Inequality

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

2.1

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

Reversing the signs of both sides yields

2.2

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

3 Third Inequality

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

3.1

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

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

and divide by $v^n$ to obtain

Taking the reciprocal, we obtain

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

3.2

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

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

and divide by $u^n$ to obtain

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

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

and the positivity condition mentioned above becomes .

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