§7.8 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); Crstici and Tudor (1975) for (7.8.7). See also Wu (1982).
Keywords:: Mill’s ratio for complementary error function, error functions, inequalities
Permalink:: http://dlmf.nist.gov/7.8

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

7.8.1 $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)=\frac{\int _{x}^{\infty}e^{{-t^{2}}}dt}{e^{{-x^{2}}}}=e^{{x^{2}}}\int _{x}^{\infty}e^{{-t^{2}}}dt.$

Defines:: $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)$ : Mills’ ratio
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $x$ : real variable
Permalink:: http://dlmf.nist.gov/7.8.E1
Encodings:: TeX, pMML, png

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

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:: $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)$ : Mills’ ratio and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E3
Encodings:: TeX, pMML, png

7.8.4 $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}},$ $x>-\tfrac{1}{2}\sqrt{2}$ ,

Symbols:: $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)$ : Mills’ ratio and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E4
Encodings:: TeX, pMML, png

7.8.5 $\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)<\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^{2}+1}{2x^{2}+3},$ $x\geq 0$ .

Symbols:: $\mathop{\mathsf{M}\/}\nolimits\!\left(x\right)$ : Mills’ ratio and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E5
Encodings:: TeX, pMML, png

Next,

7.8.6 $\int _{0}^{x}e^{{at^{2}}}dt<\frac{1}{3ax}\left(2e^{{ax^{2}}}+ax^{2}-2\right),$ $a,x>0$ .

Symbols:: $dx$ : differential of $x$ , $e$ : 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

7.8.7 $\int _{0}^{x}e^{{t^{2}}}dt<\frac{e^{{x^{2}}}-1}{x},$ $x>0$ .

Symbols:: $dx$ : differential of $x$ , $e$ : 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