Bernoulli's inequality

Bernoulli's inequality
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants:
Integer exponent
Case 1: for every integer and real number . The inequality is strict if and .
Case 2: for every integer and every real number .
Case 3: for every even integer and every real number .
Real exponent
for every real number and . The inequality is strict if and .
for every real number and .
History
Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often.
According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963), p. 177, the inequality is actually due to Sluse in his Mesolabum (1668 edition), Chapter IV "De maximis & minimis".
Proof for integer exponent
The first case has a simple inductive proof:
Suppose the statement is true for :
Then it follows that
Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form:
we prove the inequality for ,
from validity for some r we deduce validity for .
For ,
is equivalent to which is true.
Similarly, for we have
Now suppose the statement is true for :
Then it follows that
since as well as . By the modified induction we conclude the statement is true for every non-negative integer .
By noting that if , then is negative gives case 3.
Generalizations
Generalization of exponent
The exponent can be generalized to an arbitrary real number as follows: if , then
for or , and
for .
This generalization can be proved by comparing derivatives. The strict versions of these inequalities require and .
Generalization of base
Instead of the inequality holds also in the form where are real numbers, all greater than , all with the same sign. Bernoulli's inequality is a special case when . This generalized inequality can be proved by mathematical induction.
In the first step we take . In this case the inequality is obviously true.
In the second step we assume validity of the inequality for numbers and deduce validity for numbers.
We assume thatis valid. After multiplying both sides with a positive number we get:
As all have the same sign, the products are all positive numbers. So the quantity on the right-hand side can be bounded as follows:what was to be shown.
Related inequalities
The following inequality estimates the -th power of from the other side. For any real numbers and with , one has
where 2.718.... This may be proved using the inequality .
Alternative form
An alternative form of Bernoulli's inequality for and is:
This can be proved (for any integer ) by using the formula for geometric series: (using )
or equivalently
Alternative proofs
Arithmetic and geometric means
An elementary proof for and x ≥ -1 can be given using weighted AM-GM.
Let be two non-negative real constants. By weighted AM-GM on with weights respectively, we get
Note that
and
so our inequality is equivalent to
After substituting (bearing in mind that this implies ) our inequality turns into
which is Bernoulli's inequality.
Geometric series
Bernoulli's inequality
is equivalent to
and by the formula for geometric series (using y = 1 + x) we get
which leads to
Now if then by monotony of the powers each summand , and therefore their sum is greater and hence the product on the LHS of.
If then by the same arguments and thus
all addends are non-positive and hence so is their sum. Since the product of two non-positive numbers is non-negative, we get again.
Binomial theorem
One can prove Bernoulli's inequality for x ≥ 0 using the binomial theorem. It is true trivially for r = 0, so suppose r is a positive integer. Then Clearly and hence as required.
Using convexity
For the function is strictly convex. Therefore for holds
and the reversed inequality is valid for and .
Notes
References
External links
Bernoulli Inequality by Chris Boucher, Wolfram Demonstrations Project.
Inequalities