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