7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.7 Integral Representations 7.9 Continued Fractions

§7.8 Inequalities

ⓘ

Keywords:: Mill’s ratio for complementary error function, error functions, inequalities
Notes:: See Mills (1926), Mitrinović (1970, p. 177), and Gautschi (1959b) for (7.8.2); Kesavan and Vasudevamurthy (1985) for the lower bound in (7.8.3); Laforgia and Sismondi (1988) for the upper bound in (7.8.3); Gupta (1970) for (7.8.4); Luke (1969b, p. 201) for (7.8.5); Martić (1978) for (7.8.6); Janssen (2021) for (7.7.7). See also Wu (1982).
Referenced by:: Erratum (V1.0.17) for References, Erratum (V1.0.17) for Equation (7.8.8)
Permalink:: http://dlmf.nist.gov/7.8
Addition (effective with 1.0.17):: A new inequality (7.8.8) was added. This is Pólya (1949, (1.5)).
Suggested by Roberto Iacono
See also:: Annotations for Ch.7

Let $\mathsf{M}\left(x\right)$ denote Mills’ ratio:

7.8.1		$\mathsf{M}\left(x\right)=\frac{\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t}{e^{-x% ^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t.$
ⓘ Defines: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $x$ : real variable Permalink: http://dlmf.nist.gov/7.8.E1 Encodings: TeX, pMML, png See also: Annotations for §7.8 and Ch.7

(Other notations are often used.) Then

7.8.2		$\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}+% (4/\pi)}},$
$x\geq 0$ ,
ⓘ Symbols: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real variable A&S Ref: 7.1.13 Referenced by: §7.8 Permalink: http://dlmf.nist.gov/7.8.E2 Encodings: TeX, pMML, png See also: Annotations for §7.8 and Ch.7

7.8.3		$\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}\left(x\right)<\frac{1}{x+1},$
$x\geq 0$ ,
ⓘ Symbols: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real variable Referenced by: §7.8 Permalink: http://dlmf.nist.gov/7.8.E3 Encodings: TeX, pMML, png See also: Annotations for §7.8 and Ch.7

7.8.4		$\mathsf{M}\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}},$
$x>-\tfrac{1}{2}\sqrt{2}$ ,
ⓘ Symbols: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio and $x$ : real variable Referenced by: §7.8 Permalink: http://dlmf.nist.gov/7.8.E4 Encodings: TeX, pMML, png See also: Annotations for §7.8 and Ch.7

7.8.5		$\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x% \mathsf{M}\left(x\right)<\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^{2}% +1}{2x^{2}+3},$
$x\geq 0$ .
ⓘ Symbols: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio and $x$ : real variable Referenced by: §7.8 Permalink: http://dlmf.nist.gov/7.8.E5 Encodings: TeX, pMML, png See also: Annotations for §7.8 and Ch.7

Next,

7.8.6		$\int_{0}^{x}e^{at^{2}}\,\mathrm{d}t<\frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2% \right),$
$a,x>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $x$ : real variable Referenced by: §7.8 Permalink: http://dlmf.nist.gov/7.8.E6 Encodings: TeX, pMML, png See also: Annotations for §7.8 and Ch.7

7.8.7		$\frac{\sinh x^{2}}{x}<{\mathrm{e}}^{x^{2}}F\left(x\right)=\int_{0}^{x}{\mathrm% {e}}^{t^{2}}\,\mathrm{d}t<\frac{{\mathrm{e}}^{x^{2}}-1}{x},$
$x>0$ .
ⓘ Symbols: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable Referenced by: Erratum (V1.1.5) for Additions Permalink: http://dlmf.nist.gov/7.8.E7 Encodings: TeX, pMML, png Addition (effective with 1.1.5): A new inequality was added. See also: Annotations for §7.8 and Ch.7

The function $F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^{2}}}$ is strictly decreasing for $x>0$ . For these and similar results for Dawson’s integral $F\left(x\right)$ see Janssen (2021).

7.8.8		$\operatorname{erf}x<\sqrt{1-{\mathrm{e}}^{-4x^{2}/\pi}},$
$x>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable Source: Pólya (1949, (1.5)) Referenced by: §7.8, Erratum (V1.0.17) for Equation (7.8.8) Permalink: http://dlmf.nist.gov/7.8.E8 Encodings: TeX, pMML, png Addition (effective with 1.0.17): This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11). Suggested by Roberto Iacono See also: Annotations for §7.8 and Ch.7

7.7 Integral Representations 7.9 Continued Fractions

NIST

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