Hadamard inequality for determinants

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1: 1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(i) Determinants: Elementary Properties

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Relationships Between Determinants

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Hadamard’s Inequality

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§1.3(ii) Special Determinants

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Cauchy Determinant

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2: Bibliography P

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An inequality for the Bessel function $J_{\nu }(\nu x)$ . SIAM J. Math. Anal. 15 (1), pp. 203–205.

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External Links:

ISSN 0036-1410, Document, MathReview (L. Debnath)

Referenced by:

§10.14, §10.37.

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On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.

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ISSN 1364-5021, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.40(iv), §2.11(v).

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On the use of Hadamard expansions in hyperasymptotic evaluation. II. Complex variables. Proc. Roy. Soc. London Ser. A 457, pp. 2855–2869.

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External Links:

ISSN 1364-5021, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.40(iv), §2.11(v).

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Inequalities for the zeros of the Airy functions. SIAM J. Math. Anal. 22 (1), pp. 260–267.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§9.9(iv).

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3: Bibliography H

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Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).

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Notes:

Reprinted in Oeuvres de Jacques Hadamard. Tomes I, pp. 189–210, Éditions du Centre National de la Recherche Scientifique, Paris, 1968.

External Links:

ISSN 0037-9484, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§25.10(i), §27.2(i).

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Inequalities. 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1952 edition and reprinted in 1988.

External Links:

ISBN 0-521-35880-9, MathReview, ZentralBlatt

Referenced by:

§1.2(iv), §1.4(viii), §1.7(i), §1.7(ii), §1.7(iii), §1.7(iv), §4.5(i), §4.5(ii).

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4: 1.2 Elementary Algebra

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Inequalities

… ►we have Hölder’s Inequality …The triangle inequality, … ►

The Determinant

►The matrix $𝐀$ has a determinant, $det(𝐀)$, explored further in §1.3, denoted, in full index form, as …

5: 1.7 Inequalities

§1.7 Inequalities

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Cauchy–Schwarz Inequality

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Minkowski’s Inequality

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Cauchy–Schwarz Inequality

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§1.7(iv) Jensen’s Inequality

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6: Edward Neuman

… ►Neuman has published several papers on approximations and expansions, special functions, and mathematical inequalities. …

7: Bibliography

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Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

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Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt

Referenced by:

§19.9(i).

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On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§8.10.

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Gamma function inequalities. Numer. Algorithms 49 (1-4), pp. 53–84.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

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Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.

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External Links:

ISSN 0350-1302, MathReview, ZentralBlatt

Referenced by:

§19.9(i).

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8: 18.2 General Orthogonal Polynomials

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§18.2(ix) Moments

… ►The Hankel determinant ${\mathrm{\Delta }}_n$ of order $n$ is defined by ${\mathrm{\Delta }}_0=1$ and …The recurrence coefficients ${\alpha }_n$ and ${\beta }_n$ in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by …It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. … ► …

9: 4.32 Inequalities

§4.32 Inequalities

… ►For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).

10: 25.10 Zeros

… ►Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). …