Hadamard inequality

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11: 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

12: 24.9 Inequalities

§24.9 Inequalities

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

13: 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.) …

14: 4.18 Inequalities

§4.18 Inequalities

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Jordan’s Inequality

… ►For more inequalities see Mitrinović (1964, pp. 101–111), Mitrinović (1970, pp. 235–265), and Bullen (1998, pp. 250–254).

15: 5.6 Inequalities

§5.6 Inequalities

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Gautschi’s Inequality

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Kershaw’s Inequality

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16: 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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17: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. …

18: 10.14 Inequalities; Monotonicity

§10.14 Inequalities; Monotonicity

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Kapteyn’s Inequality

… ►For inequalities for the function $\mathrm{\Gamma }\left(\nu +1\right){(2/x)}^{\nu }{J}_{\nu }\left(x\right)$ with $\nu >-\frac{1}{2}$ see Neuman (2004). …

19: 13.22 Zeros

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20: Richard A. Askey

… ►Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. …This inequality was a key element of Louis de Branges’ proof of the Bieberbach conjecture in 1985. …