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

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

Title inequalities for differences of powers InequalitiesForDifferencesOfPowers 2013-03-22 15:48:42 2013-03-22 15:48:42 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Theorem msc 26D99 BernoullisInequality