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

In real analysis, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x.

The inequality states that

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

for every integer r ≥ 2 and every real number x ≥ −1 with x ≠ 0.

There is also a generalized version that says for every real number r ≥ 1 and real number x ≥ -1,

while for 0 ≤ r ≤ 1 and real number x ≥ -1,

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.

Contents

HistoryEdit

Jacob Bernoulli first published the inequality in his treatise “Positiones Arithmeticae de Seriebus Infinitis” (Basel, 1689), where he used the inequality often.[1]

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".[1]

Proof of the inequalityEdit

For r = 0,

 

is equivalent to 1 ≥ 1 which is true as required.

Now suppose the statement is true for r = k:

 

Then it follows that

 

By induction we conclude the statement is true for all r ≥ 0.

GeneralizationEdit

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

 

for r ≤ 0 or r ≥ 1, and

 

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 inequalitiesEdit

The following inequality estimates the r-th power of 1 + x from the other side. For any real numbers xr  with r > 0, one has

 

where e = 2.718.... This may be proved using the inequality (1 + 1/k)k < e.

Alternative formEdit

An alternative form of Bernoulli's inequality for   and   is:

 

This can be proved (for integer t) by using the formula for geometric series: (using y=1-x)

 

or equivalently  

Alternative ProofEdit

Using AM-GM

An elementary proof for   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.

Using the formula for geometric series

Bernoulli's inequality

 

 

 

 

 

(1)

is equal 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  , 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 their sum. Since the product of two non-positive numbers is non-negative, we get again (4), which proves Bernoulli's inequality even for  .

Using Binomial theorem

(1) For x > 0,   Obviously,  

Thus,  

(2) For x = 0,  

(3) For −1 ≤ x < 0, let y = −x, then 0 < y ≤ 1

Replace x with −y, we have  

Also, according to the binomial theorem,  

then 

Notice that  

Therefore, we can see that each binomial term   is multiplied by a factor   , and that will make each term smaller than the term before.

For that reason,  

Hence,  

Replace y with −x back, we get  

Notice that by using binomial theorem, we can only prove the cases when r is a positive integer or zero.

NotesEdit

ReferencesEdit

  • Carothers, N.L. (2000). Real analysis. Cambridge: Cambridge University Press. p. 9. ISBN 978-0-521-49756-5.
  • Bullen, P. S. (2003). Handbook of means and their inequalities. Dordercht [u.a.]: Kluwer Academic Publ. p. 4. ISBN 978-1-4020-1522-9.
  • Zaidman, S. (1997). Advanced calculus : an introduction to mathematical analysis. River Edge, NJ: World Scientific. p. 32. ISBN 978-981-02-2704-3.

External linksEdit