§4.5 Inequalities

§4.5(i) Logarithms

 4.5.1 $\frac{x}{1+x}<\ln\left(1+x\right) $x>-1$, $x\neq 0$, ⓘ 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, ⓘ 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

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 $x<1$, ⓘ 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 $-\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} $x>-1$, ⓘ 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 $x<1$, ⓘ 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} $x>0$, $y>0$, ⓘ 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, ⓘ 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).