Jordan inequality

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11: 1.9 Calculus of a Complex Variable

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Triangle Inequality

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Jordan Curve Theorem

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12: 10.37 Inequalities; Monotonicity

§10.37 Inequalities; Monotonicity

… ►For sharper inequalities when the variables are real see Paris (1984) and Laforgia (1991). …

13: 18.14 Inequalities

§18.14 Inequalities

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Legendre

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Jacobi

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Szegő–Szász Inequality

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14: 7.8 Inequalities

§7.8 Inequalities

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7.8.7 $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

Referenced by:

Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/7.8.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

A new inequality was added.

See also:

Annotations for §7.8 and Ch.7

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7.8.8 $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7

15: 27.3 Multiplicative Properties

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27.3.4 $$J_k\left(n\right)=n^k\prod _{p|n}(1-p^{-k}),$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $k$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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16: 27.2 Functions

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27.2.11 $$J_k\left(n\right)=\sum _{\left(\left(d_1,\mathrm{\dots },d_k\right),n\right)=1}1,$$

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Defines:

$J_k\left(n\right)$: Jordan’s function

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Referenced by:

§27.2(i)

Permalink:

http://dlmf.nist.gov/27.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►This is Jordan’s function. Note that $J_1\left(n\right)=\varphi \left(n\right)$. …

17: 24.9 Inequalities

§24.9 Inequalities

►Except where otherwise noted, the inequalities in this section hold for $n=1,2,\mathrm{\dots }$. …

18: 4.5 Inequalities

§4.5 Inequalities

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§4.5(i) Logarithms

… ►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

… ►(When $x=0$ the inequalities become equalities.) …

19: 5.6 Inequalities

§5.6 Inequalities

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Gautschi’s Inequality

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Kershaw’s Inequality

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20: 19.24 Inequalities

§19.24 Inequalities

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§19.24(i) Complete Integrals

… ►Other inequalities can be obtained by applying Carlson (1966, Theorems 2 and 3) to (19.16.20)–(19.16.23). … ► ►

§19.24(ii) Incomplete Integrals

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