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:[1]

An illustration of Bernoulli's inequality, with the graphs of and shown in red and blue respectively. Here,

Integer exponent

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  • Case 1:   for every integer   and real number  . The inequality is strict if   and  .
  • Case 2:   for every integer   and every real number  .[2]
  • Case 3:   for every even integer   and every real number  .

Real exponent

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  •   for every real number   and  . The inequality is strict if   and  .
  •   for every real number   and  .

History

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Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often.[3]

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".[3]

Proof for integer exponent

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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

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Generalization of exponent

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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

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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.

Proof

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 that is 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.

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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

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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

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Arithmetic and geometric means

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An elementary proof for   and   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

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Bernoulli's inequality

  (1)

is equivalent to

  (2)

and by the formula for geometric series (using y = 1 + x) we get

  (3)

which leads to

  (4)

Now if   then by monotony of the powers each summand  , and therefore their sum is greater   and hence the product on the LHS of (4).

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 (4).

Binomial theorem

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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

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For   the function   is strictly convex. Therefore, for   holds   and the reversed inequality is valid for   and  .

Another way of using convexity is to re-cast the desired inequality to   for real   and real  . This inequality can be proved using the fact that the   function is concave, and then using Jensen's inequality in the form   to give:   which is the desired inequality.

Notes

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  1. ^ Brannan, D. A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 20. ISBN 9781139458955.
  2. ^ Excluding the case r = 0 and x = –1, or assuming that 00 = 1.
  3. ^ a b mathematics – First use of Bernoulli's inequality and its name – History of Science and Mathematics Stack Exchange

References

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