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Hadamard inequality for determinants

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1: 1.3 Determinants, Linear Operators, and Spectral Expansions
§1.3(i) Determinants: Elementary Properties
Relationships Between Determinants
Hadamard’s Inequality
§1.3(ii) Special Determinants
Cauchy Determinant
2: Bibliography P
  • R. B. Paris (1984) An inequality for the Bessel function J ν ( ν x ) . SIAM J. Math. Anal. 15 (1), pp. 203–205.
  • R. B. Paris (2001a) On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.
  • R. B. Paris (2001b) On the use of Hadamard expansions in hyperasymptotic evaluation. II. Complex variables. Proc. Roy. Soc. London Ser. A 457, pp. 2855–2869.
  • G. Pittaluga and L. Sacripante (1991) Inequalities for the zeros of the Airy functions. SIAM J. Math. Anal. 22 (1), pp. 260–267.
  • 3: Bibliography H
  • J. Hadamard (1896) Sur la distribution des zéros de la fonction ζ ( s ) et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).
  • G. H. Hardy, J. E. Littlewood, and G. Pólya (1967) Inequalities. 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge.
  • 4: 1.2 Elementary Algebra
    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
    Cauchy–Schwarz Inequality
    Minkowski’s Inequality
    Cauchy–Schwarz Inequality
    §1.7(iv) Jensen’s Inequality
    6: Edward Neuman
    Neuman has published several papers on approximations and expansions, special functions, and mathematical inequalities. …
    7: Bibliography
  • W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.
  • H. Alzer and S. Qiu (2004) Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.
  • H. Alzer (1997b) On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.
  • H. Alzer (2008) Gamma function inequalities. Numer. Algorithms 49 (1-4), pp. 53–84.
  • G. D. Anderson and M. K. Vamanamurthy (1985) Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.
  • 8: 18.2 General Orthogonal Polynomials
    §18.2(ix) Moments
    The Hankel determinant Δ n of order n is defined by Δ 0 = 1 and …The recurrence coefficients α n and β 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, ζ ( s ) 0 for 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). …