4 Elementary Functions Logarithm, Exponential, Powers 4.4 Special Values and Limits 4.6 Power Series

§4.5 Inequalities

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Permalink:: http://dlmf.nist.gov/4.5
See also:: Annotations for Ch.4

§4.5(i) Logarithms
§4.5(ii) Exponentials

§4.5(i) Logarithms

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Keywords:: inequalities, logarithm function
Notes:: (4.5.1) and (4.5.5) can be verified by the methods of Hardy et al. (1967, pp. 106–107). (4.5.2) and (4.5.4) follow from (4.5.1). (4.5.3) follows from the fact that $x=0$ and $x=0.5828\dots$ are successive zeros of $\frac{3}{2}x+\ln\left(1-x\right)$ . (4.5.6) is obtained from the Maclaurin expansion of $\ln\left(1+z\right)$ .
Permalink:: http://dlmf.nist.gov/4.5.i
See also:: Annotations for §4.5 and Ch.4

4.5.1		$\frac{x}{1+x}<\ln\left(1+x\right)<x,$
$x>-1$ , $x\neq 0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.1.33 Referenced by: §4.5(i), §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E1 Encodings: TeX, pMML, png See also: Annotations for §4.5(i), §4.5 and Ch.4

4.5.2		$x<-\ln\left(1-x\right)<\frac{x}{1-x},$
$x<1$ , $x\neq 0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.1.34 Referenced by: §4.5(i), §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E2 Encodings: TeX, pMML, png See also: Annotations for §4.5(i), §4.5 and Ch.4

4.5.3		$\|\ln\left(1-x\right)\|<\tfrac{3}{2}x,$
$0<x\leq 0.5828\dots$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.1.35 Referenced by: §4.5(i) Permalink: http://dlmf.nist.gov/4.5.E3 Encodings: TeX, pMML, png See also: Annotations for §4.5(i), §4.5 and Ch.4

4.5.4		$\ln x\leq x-1,$
$x>0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.1.36 Referenced by: §4.5(i) Permalink: http://dlmf.nist.gov/4.5.E4 Encodings: TeX, pMML, png See also: Annotations for §4.5(i), §4.5 and Ch.4

4.5.5		$\ln x\leq a(x^{1/a}-1),$
$a$ , $x>0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $a$ : real or complex constant and $x$ : real variable A&S Ref: 4.1.37 Referenced by: §4.5(i) Permalink: http://dlmf.nist.gov/4.5.E5 Encodings: TeX, pMML, png See also: Annotations for §4.5(i), §4.5 and Ch.4

4.5.6		$\|\ln\left(1+z\right)\|\leq-\ln\left(1-\|z\|\right),$
$\|z\|<1$ .
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 4.1.38 Referenced by: §4.5(i) Permalink: http://dlmf.nist.gov/4.5.E6 Encodings: TeX, pMML, png See also: Annotations for §4.5(i), §4.5 and Ch.4

For more inequalities involving the logarithm function see Mitrinović (1964, pp. 75–77), Mitrinović (1970, pp. 272–276), and Bullen (1998, pp. 159–160).

§4.5(ii) Exponentials

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Keywords:: exponential function, inequalities
Notes:: (4.5.7) to (4.5.12) are obtained by exponentiating the inequalities (4.5.1) and (4.5.2). For (4.5.13), see Hardy et al. (1967, p. 102). (4.5.14) follows from the fact that $1-\tfrac{1}{2}x-e^{-x}$ has $0$ and $1.5936\dots$ as consecutive zeros. (4.5.15) and (4.5.16) can be derived from the Maclaurin expansion of $e^{z}$ .
Permalink:: http://dlmf.nist.gov/4.5.ii
See also:: Annotations for §4.5 and Ch.4

In (4.5.7)–(4.5.12) it is assumed that $x\neq 0$ . (When $x=0$ the inequalities become equalities.)

4.5.7		$e^{-x/(1-x)}<1-x<e^{-x},$
$x<1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.29 Referenced by: §4.5(ii), §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E7 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.8		$1+x<e^{x},$
$-\infty<x<\infty$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.30 Permalink: http://dlmf.nist.gov/4.5.E8 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.9		$e^{x}<\frac{1}{1-x},$
$x<1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.31 Permalink: http://dlmf.nist.gov/4.5.E9 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.10		$\frac{x}{1+x}<1-e^{-x}<x,$
$x>-1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.32 Permalink: http://dlmf.nist.gov/4.5.E10 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.11		$x<e^{x}-1<\frac{x}{1-x},$
$x<1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.33 Permalink: http://dlmf.nist.gov/4.5.E11 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.12		$e^{x/(1+x)}<1+x,$
$x>-1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.34 Referenced by: §4.5(ii), §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E12 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.13		$e^{xy/(x+y)}<\left(1+\frac{x}{y}\right)^{y}<e^{x},$
$x>0$ , $y>0$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $x$ : real variable and $y$ : real variable A&S Ref: 4.2.36 Referenced by: §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E13 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.14		$e^{-x}<1-\tfrac{1}{2}x,$
$0<x\leq 1.5936\dots$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $x$ : real variable A&S Ref: 4.2.37 Referenced by: §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E14 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.15		$\tfrac{1}{4}\|z\|<\|e^{z}-1\|<\tfrac{7}{4}\|z\|,$
$0<\|z\|<1$ ,
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable A&S Ref: 4.2.38 Referenced by: §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E15 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

4.5.16		$\|e^{z}-1\|\leq e^{\|z\|}-1\leq\|z\|e^{\|z\|},$
$z\in\mathbb{C}$ .
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable A&S Ref: 4.2.39 Referenced by: §4.5(ii) Permalink: http://dlmf.nist.gov/4.5.E16 Encodings: TeX, pMML, png See also: Annotations for §4.5(ii), §4.5 and Ch.4

For more inequalities involving the exponential function see Mitrinović (1964, pp. 73–77), Mitrinović (1970, pp. 266–271), and Bullen (1998, pp. 81–83).

4.4 Special Values and Limits 4.6 Power Series

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§4.5 Inequalities

Contents

§4.5(i) Logarithms

§4.5(ii) Exponentials