Digital Library of Mathematical Functions
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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.5 Inequalities

Contents

§4.5(i) Logarithms

4.5.1 x1+x<ln(1+x)<x,
x>-1, x0,
4.5.2 x<-ln(1-x)<x1-x,
x<1, x0,
4.5.3 |ln(1-x)|<32x,
0<x0.5828,
4.5.4 lnxx-1,
x>0,
4.5.5 lnxa(x1/a-1),
a, x>0,
4.5.6 |ln(1+z)|-ln(1-|z|),
|z|<1.

For more inequalities involving the logarithm function see Mitrinović (1964, pp. 75–77), Mitrinović (1970, pp. 272–276), and Bullen (1998, pp. 159–160).

§4.5(ii) Exponentials

In (4.5.7)–(4.5.12) it is assumed that x0. (When x=0 the inequalities become equalities.)

4.5.7 -x/(1-x)<1-x<-x,
x<1,
4.5.8 1+x<x,
-<x<,
4.5.9 x<11-x,
x<1,
4.5.10 x1+x<1--x<x,
x>-1,
4.5.11 x<x-1<x1-x,
x<1,
4.5.12 x/(1+x)<1+x,
x>-1,
4.5.13 xy/(x+y)<(1+xy)y<x,
x>0, y>0,
4.5.14 -x<1-12x,
0<x1.5936,
4.5.15 14|z|<|z-1|<74|z|,
0<|z|<1,
4.5.16 |z-1||z|-1|z||z|,
z.

For more inequalities involving the exponential function see Mitrinović (1964, pp. 73–77), Mitrinović (1970, pp. 266–271), and Bullen (1998, pp. 81–83).