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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.8 Inequalities

Let M(x) denote Mills’ ratio:

7.8.1 M(x)=xet2dtex2=ex2xet2dt.

(Other notations are often used.) Then

7.8.2 1x+x2+2<M(x)1x+x2+(4/π),
x0,
7.8.3 π2πx+2M(x)<1x+1,
x0,
7.8.4 M(x)<23x+x2+4,
x>122,
7.8.5 x22x2+1x2(2x2+5)4x4+12x2+3xM(x)<2x4+9x2+44x4+20x2+15<x2+12x2+3,
x0.

Next,

7.8.6 0xeat2dt<13ax(2eax2+ax22),
a,x>0.
7.8.7 sinhx2x<ex2F(x)=0xet2dt<ex21x,
x>0.

The function F(x)/1e2x2 is strictly decreasing for x>0. For these and similar results for Dawson’s integral F(x) see Janssen (2021).

7.8.8 erfx<1e4x2/π,
x>0.