# §7.8 Inequalities

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

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

(Other notations are often used.) Then

 7.8.2 $\frac{1}{x+\sqrt{x^{2}+2}}<\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)\leq% \frac{1}{x+\sqrt{x^{2}+(4/\pi)}},$ $x\geq 0$, Symbols: $\mathop{\mathsf{M}\/}\nolimits\!\left(\NVar{x}\right)$: 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
 7.8.3 $\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathop{\mathsf{M}\/}\nolimits\!\left(x% \right)<\frac{1}{x+1},$ $x\geq 0$, Symbols: $\mathop{\mathsf{M}\/}\nolimits\!\left(\NVar{x}\right)$: 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
 7.8.4 $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}},$ $x>-\tfrac{1}{2}\sqrt{2}$, Symbols: $\mathop{\mathsf{M}\/}\nolimits\!\left(\NVar{x}\right)$: 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
 7.8.5 $\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x% \mathop{\mathsf{M}\/}\nolimits\!\left(x\right)<\frac{2x^{4}+9x^{2}+4}{4x^{4}+2% 0x^{2}+15}<\frac{x^{2}+1}{2x^{2}+3},$ $x\geq 0$. Symbols: $\mathop{\mathsf{M}\/}\nolimits\!\left(\NVar{x}\right)$: 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

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 exponential function, $\int$: 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
 7.8.7 $\int_{0}^{x}e^{t^{2}}\mathrm{d}t<\frac{e^{x^{2}}-1}{x},$ $x>0$. Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral and $x$: real variable Referenced by: §7.8 Permalink: http://dlmf.nist.gov/7.8.E7 Encodings: TeX, pMML, png See also: Annotations for 7.8