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

§7.8 Inequalities

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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 $\mathsf{M}\left(x\right)=\frac{\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t}{e^{-x% ^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t.$

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Defines:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : 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 $\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}+% (4/\pi)}},$ $x\geq 0$ ,

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Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and : 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 $\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}\left(x\right)<\frac{1}{x+1},$ $x\geq 0$ ,

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Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and : 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 $\mathsf{M}\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}},$ $x>-\tfrac{1}{2}\sqrt{2}$ ,

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Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio and : 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 $\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x% \mathsf{M}\left(x\right)<\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^{2}% +1}{2x^{2}+3},$ $x\geq 0$ .

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Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio and : 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 $\int_{0}^{x}e^{at^{2}}\,\mathrm{d}t<\frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2% \right),$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : 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 $\frac{\sinh x^{2}}{x}<{\mathrm{e}}^{x^{2}}F\left(x\right)=\int_{0}^{x}{\mathrm% {e}}^{t^{2}}\,\mathrm{d}t<\frac{{\mathrm{e}}^{x^{2}}-1}{x},$

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Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and : 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 . For these and similar results for Dawson’s integral $F\left(x\right)$ see Janssen (2021).

7.8.8 $\operatorname{erf}x<\sqrt{1-{\mathrm{e}}^{-4x^{2}/\pi}},$

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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 : 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

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