# §7.8 Inequalities

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