Neper's inequality

The Nepersche inequality ( English Napier’s inequality ) is an inequality of the mathematical branch of analysis , which goes back to the Scottish mathematician John Napier (1550-1617). It provides elementary lower and upper estimates for the real natural logarithm .

Representation of the inequality

The inequality is as follows:

Let two real numbers be given and let it hold . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle 0<a<b}$

Then the inequalities exist

(N)  . ${\displaystyle {\frac {1}{b}}<{\frac {\ln b-\ln a}{b-a}}<{\frac {1}{a}}}$

Derivation of the nepersian inequality by means of integral calculus

Equivalent to the Nepersian inequality is the following:

(N ^' )  . ${\displaystyle {\frac {b-a}{b}}<\ln b-\ln a<{\frac {b-a}{a}}}$

So one gets the Nepersian inequality by means of integral calculus . According to this, the middle term of (N ^' ) is nothing more than the content of the area below the function graph of the real reciprocal function in the interval . ${\displaystyle [a,b]}$

application

A useful application of the Nepersian inequality results if one still bets in it as well as - for a natural number - . ${\displaystyle a=1}$ ${\displaystyle n}$ ${\displaystyle b=1+{\tfrac {1}{n}}}$

Then it results because of and ${\displaystyle b-a={\tfrac {1}{n}}}$ ${\displaystyle \ln a=0}$

${\displaystyle {\frac {1}{1+{\frac {1}{n}}}}<{\frac {\ln \left(1+{\frac {1}{n}}\right)}{\frac {1}{n}}}<{\frac {1}{1}}}$

and further

${\displaystyle {\frac {n}{n+1}}<n\cdot \ln \left(1+{\frac {1}{n}}\right)<1}$

and finally

${\displaystyle {\frac {n}{n+1}}<\ln {\left(1+{\frac {1}{n}}\right)}^{n}<1}$ .

By forming a limit one then obtains

${\displaystyle \lim _{n\to \infty }{\ln {\left(1+{\frac {1}{n}}\right)}^{n}}=1}$

and it follows for reasons of continuity and by applying the exponential function

${\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\exp(1)=e}$ .

Related inequalities

The Nepean inequality can be sharpened considerably. This is shown, for example, by the inequality of Hermite-Hadamard , which leads to the Neper inequality. Because if one takes into account the fact that the restriction of the real inverse function to the interval of positive numbers is a convex function , then the estimations result immediately ${\displaystyle ]0,\infty [}$ ${\displaystyle 0<a<b}$

${\displaystyle {\frac {1}{\left({\tfrac {a+b}{2}}\right)}}\leq {\frac {\ln b-\ln a}{b-a}}\leq {\frac {{\frac {1}{a}}+{\frac {1}{b}}}{2}}}$

and thus

${\displaystyle {\frac {2}{a+b}}\leq {\frac {\ln b-\ln a}{b-a}}\leq {\frac {1}{2}}\left({\frac {1}{a}}+{\frac {1}{b}}\right)}$ .

In the event that is particular , one even has the following - and better ones in this case! - Estimates: ${\displaystyle b=a+1}$

${\displaystyle {\frac {2}{2a+1}}<\ln \left(1+{\frac {1}{a}}\right)<{\frac {1}{\sqrt {a^{2}+a}}}}$ .

literature

• Claudi Alsina , Roger B. Nelsen : Math Made Visual: Creating Images for Understanding Mathematics (=  Classroom Resource Materials Series ). The Mathematical Association of America , Washington, DC 2006, ISBN 0-88385-746-4 ( MR2216733 ).
• DS Mitrinović : Analytic Inequalities . In cooperation with PM Vasić (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 165 ). Springer Verlag , Berlin ( inter alia ) 1970, ISBN 3-540-62903-3 ( MR0274686 ).

Individual evidence

1. ^ ^{A b} Claudi Alsina, Roger B. Nelsen: Math Made Visual: Creating Images for Understanding Mathematics. 2006, p. 16
2. ↑ The front inequality, even if formulated for the reciprocal values, can be found in: DS Mitrinović: Analytic Inequalities. 1970, p. 273
3. ↑ DS Mitrinović: Analytic Inequalities. 1970, pp. 273-274