Definitions

Bernoulli's inequality

In real analysis, Bernoulli's inequality is an inequality that approximates exponentiations of 1 + x.

The inequality states that

$\left(1 + x\right)^r geq 1 + rx!$
for every integer r ≥ 0 and every real number x > −1. If the exponent r is even, then the inequality is valid for all real numbers x. The strict version of the inequality reads
$\left(1 + x\right)^r > 1 + rx!$
for every integer r ≥ 2 and every real number x ≥ −1 with x ≠ 0.

Bernoulli's inequality is often used as the crucial step in the proof of other inequalities. It can itself be proved using mathematical induction, as shown below.

Proof of the inequality

For $r=0,,$

$\left(1+x\right)^0 ge 1+0x$
is equivalent to $1ge 1$ which is true as required.

Now suppose the statement is true for $r=k$:

$\left(1+x\right)^k ge 1+kx.$
Then it follows that
$\left(1+x\right)\left(1+x\right)^k ge \left(1+x\right)\left(1+kx\right)$ (by hypothesis, since $\left(1+x\right)ge 0$)

$begin\left\{matrix\right\}$
& iff & (1+x)^{k+1} ge 1+kx+x+kx^2 & iff & (1+x)^{k+1} ge 1+(k+1)x+kx^2 end{matrix}.

However, as $1+\left(k+1\right)x + kx^2 ge 1+\left(k+1\right)x$ (since $kx^2 ge 0$), it follows that $\left(1+x\right)^\left\{k+1\right\} ge 1+\left(k+1\right)x$, which means the statement is true for $r=k+1$ as required.

By induction we conclude the statement is true for all $rge 0.$

Generalization

The exponent r can be generalized to an arbitrary real number as follows: if x > −1, then

$\left(1 + x\right)^r geq 1 + rx!$
for r ≤ 0 or r ≥ 1, and
$\left(1 + x\right)^r leq 1 + rx!$
for 0 ≤ r ≤ 1. This generalization can be proved by comparing derivatives. Again, the strict versions of these inequalities require x ≠ 0 and r ≠ 0, 1.

Related inequalities

The following inequality estimates the r-th power of 1 + x from the other side. For any real numbers x, r > 0, one has
$\left(1 + x\right)^r le e^\left\{rx\right\},!$
where e = 2.718.... This may be proved using the inequality (1 + 1/k)k < e.

References

• Carothers, N. (2000). Real Analysis. Cambridge: Cambridge University Press.
• Bullen, P.S. (1987). Handbook of Means and Their Inequalities. Berlin: Springer.
• Zaidman, Samuel (1997). Advanced Calculus. City: World Scientific Publishing Company.