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On some level of math in school we learn about Bernoulli's inequality.

Proof of its correctness is very common in textbooks as exercise, when we learn mathematical induction.

Is Bernoulli's inequality interesting just because "it looks good" and "it's good exercise for learning induction" or has it some real-world application?

Cyprian Gepfert
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  • What do you mean by "real world application"? It is used to help solve other mathematical problems, but how would there be an application of the inequality in and of itself in the real world? The sum for the first $n$ positive integers...does that have an application, really? Etc. – Daniel W. Farlow Jul 27 '16 at 16:40

1 Answers1

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This definitely won't be an easy task to provide a complete answer. I will try to figure out some points from my applications. Sorry to be messy...

  1. It might be the simplest way to utilize Bernoulli's inequality to prove the Weight-Power Summation Inequalities ($a_i,b_i>0$, $i\in\{1,\cdots,n\}$) \begin{align} \sum_{i=1}^n \frac{a_i^{m+1}}{b_i^m}&\ge\frac{\left(\sum_{i=1}^n a_i\right)^{m+1}}{\left(\sum_{i=1}^nb_i\right)^m},\;m<-1\text{ or }m>0 \end{align} or \begin{align} \sum_{i=1}^n \frac{a_i^{m+1}}{b_i^m}&\le\frac{\left(\sum_{i=1}^n a_i\right)^{m+1}}{\left(\sum_{i=1}^n b_i\right)^m},\;-1\le m\le0\label{eq:Weight-PowerSum-1<m<0} \end{align}

  2. To prove a lot of simple/important inequalities like $n-1+x^n\ge nx$ or $x^{n+1}+ny^{n+1}\ge(n+1)xy^{n}$ for $n\ge0$ and $x,y>0$,

  3. $x^y+y^x>1$ for $x,y>0$,

By chance, I found this post might be part of the answer.

Especially the generalized version to Item 2:

$$\frac{1-\frac{1}{x^n}}{n}\le x-1\le\frac{x^n-1}{n}.$$

One example for test can be shown in GeoGebra.

MathArt
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