§4.5 Inequalities

Permalink:: http://dlmf.nist.gov/4.5

§4.5(i) Logarithms
§4.5(ii) Exponentials

§4.5(i) Logarithms

Notes:: (4.5.1) and (4.5.5) can be verified by the methods of Hardy et al. (1967, pp. 106–107). (4.5.2) and (4.5.4) follow from (4.5.1). (4.5.3) follows from the fact that $x=0$ and $x=0.5828\dots$ are successive zeros of $\frac{3}{2}x+\mathop{\ln\/}\nolimits\!\left(1-x\right)$ . (4.5.6) is obtained from the Maclaurin expansion of $\mathop{\ln\/}\nolimits\!\left(1+z\right)$ .
Keywords:: inequalities, logarithm function
Permalink:: http://dlmf.nist.gov/4.5.i

4.5.1 $\frac{x}{1+x}<\mathop{\ln\/}\nolimits\!\left(1+x\right)<x,$ $x>-1$ , $x\neq 0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.1.33
Referenced by:: §4.5(i), §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E1
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4.5.2 $x<-\mathop{\ln\/}\nolimits\!\left(1-x\right)<\frac{x}{1-x},$ $x<1$ , $x\neq 0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.1.34
Referenced by:: §4.5(i), §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E2
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4.5.3 $|\mathop{\ln\/}\nolimits\!\left(1-x\right)|<\tfrac{3}{2}x,$ $0<x\leq 0.5828\dots$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.1.35
Referenced by:: §4.5(i)
Permalink:: http://dlmf.nist.gov/4.5.E3
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4.5.4 $\mathop{\ln\/}\nolimits x\leq x-1,$ $x>0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 4.1.36
Referenced by:: §4.5(i)
Permalink:: http://dlmf.nist.gov/4.5.E4
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4.5.5 $\mathop{\ln\/}\nolimits x\leq a(x^{{1/a}}-1),$ $a$ , $x>0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $a$ : real or complex constant and $x$ : real variable
A&S Ref:: 4.1.37
Referenced by:: §4.5(i)
Permalink:: http://dlmf.nist.gov/4.5.E5
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4.5.6 $|\mathop{\ln\/}\nolimits\!\left(1+z\right)|\leq-\mathop{\ln\/}\nolimits\!\left(1-|z|\right),$ $|z|<1$ .

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.1.38
Referenced by:: §4.5(i)
Permalink:: http://dlmf.nist.gov/4.5.E6
Encodings:: TeX, pMML, png

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

Notes:: (4.5.7) to (4.5.12) are obtained by exponentiating the inequalities (4.5.1) and (4.5.2). For (4.5.13), see Hardy et al. (1967, p. 102). (4.5.14) follows from the fact that $1-\tfrac{1}{2}x-e^{{-x}}$ has 0 and $1.5936\dots$ as consecutive zeros. (4.5.15) and (4.5.16) can be derived from the Maclaurin expansion of $e^{z}$ .
Keywords:: exponential function, inequalities
Permalink:: http://dlmf.nist.gov/4.5.ii

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

4.5.7 $e^{{-x/(1-x)}}<1-x<e^{{-x}},$ $x<1$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.29
Referenced by:: §4.5(ii), §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E7
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4.5.8 $1+x<e^{x},$ $-\infty<x<\infty$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.30
Permalink:: http://dlmf.nist.gov/4.5.E8
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4.5.9 $e^{x}<\frac{1}{1-x},$ $x<1$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.31
Permalink:: http://dlmf.nist.gov/4.5.E9
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4.5.10 $\frac{x}{1+x}<1-e^{{-x}}<x,$ $x>-1$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.32
Permalink:: http://dlmf.nist.gov/4.5.E10
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4.5.11 $x<e^{x}-1<\frac{x}{1-x},$ $x<1$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.33
Permalink:: http://dlmf.nist.gov/4.5.E11
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4.5.12 $e^{{x/(1+x)}}<1+x,$ $x>-1$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.34
Referenced by:: §4.5(ii), §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E12
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4.5.13 $e^{{xy/(x+y)}}<\left(1+\frac{x}{y}\right)^{y}<e^{x},$ $x>0$ , $y>0$ ,

Symbols:: $e$ : base of exponential function, $x$ : real variable and $y$ : real variable
A&S Ref:: 4.2.36
Referenced by:: §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E13
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4.5.14 $e^{{-x}}<1-\tfrac{1}{2}x,$ $0<x\leq 1.5936\dots$ ,

Symbols:: $e$ : base of exponential function and $x$ : real variable
A&S Ref:: 4.2.37
Referenced by:: §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E14
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4.5.15 $\tfrac{1}{4}|z|<|e^{z}-1|<\tfrac{7}{4}|z|,$ $0<|z|<1$ ,

Symbols:: $e$ : base of exponential function and $z$ : complex variable
A&S Ref:: 4.2.38
Referenced by:: §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E15
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4.5.16 $|e^{z}-1|\leq e^{{|z|}}-1\leq|z|e^{{|z|}},$ $z\in\Complex$ .

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $e$ : base of exponential function and $z$ : complex variable
A&S Ref:: 4.2.39
Referenced by:: §4.5(ii)
Permalink:: http://dlmf.nist.gov/4.5.E16
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For more inequalities involving the exponential function see Mitrinović (1964, pp. 73–77), Mitrinović (1970, pp. 266–271), and Bullen (1998, pp. 81–83).

§4.5 Inequalities

Contents

§4.5(i) Logarithms

§4.5(ii) Exponentials