inequalities for differences of powers

Oftentimes, one needs to estimate differences of powers of real numbers. The following inequalities are useful for this purpose:

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

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

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

Here $n$ is an integer greater than 1. The first inequality holds when $0<v<u$ , the second inequality holds when $-1<x$ , and the third inequality holds when $-1<x<1/(n-1)$ . Equality can only occur in the latter two inequalities when $x=0$ .

Title	inequalities for differences of powers
Canonical name	InequalitiesForDifferencesOfPowers
Date of creation	2013-03-22 15:48:42
Last modified on	2013-03-22 15:48:42
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	11
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 26D99
Related topic	BernoullisInequality