# §4.5 Inequalities

## §4.5(i) Logarithms

 4.5.1 $\frac{x}{1+x}<\mathop{\ln\/}\nolimits\!\left(1+x\right) $x>-1$, $x\neq 0$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.5.2 $x<-\mathop{\ln\/}\nolimits\!\left(1-x\right)<\frac{x}{1-x},$ $x<1$, $x\neq 0$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.5.3 $|\mathop{\ln\/}\nolimits\!\left(1-x\right)|<\tfrac{3}{2}x,$ $0, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.5.4 $\mathop{\ln\/}\nolimits x\leq x-1,$ $x>0$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.5.5 $\mathop{\ln\/}\nolimits x\leq a(x^{1/a}-1),$ $a$, $x>0$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.5.6 $|\mathop{\ln\/}\nolimits\!\left(1+z\right)|\leq-\mathop{\ln\/}\nolimits\!\left% (1-|z|\right),$ $|z|<1$. Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: 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)

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 exponential function 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)
 4.5.8 $1+x $-\infty, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.9 $e^{x}<\frac{1}{1-x},$ $x<1$, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.10 $\frac{x}{1+x}<1-e^{-x} $x>-1$, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.11 $x $x<1$, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.12 $e^{x/(1+x)}<1+x,$ $x>-1$, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.13 $e^{xy/(x+y)}<\left(1+\frac{x}{y}\right)^{y} $x>0$, $y>0$, Symbols: $\mathrm{e}$: base of exponential function, $x$: 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)
 4.5.14 $e^{-x}<1-\tfrac{1}{2}x,$ $0, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.15 $\tfrac{1}{4}|z|<|e^{z}-1|<\tfrac{7}{4}|z|,$ $0<|z|<1$, Symbols: $\mathrm{e}$: base of exponential function 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)
 4.5.16 $|e^{z}-1|\leq e^{|z|}-1\leq|z|e^{|z|},$ $z\in\mathbb{C}$. Symbols: $\mathbb{C}$: complex plane, $\in$: element of, $\mathrm{e}$: base of exponential function 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)

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