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As far as I remembered there is an inverse Hadamard inequality for the determinant of the form $$ |D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)} $$ providing all values in $(\cdot)>0$.

Please help me with exact references to this inequality, its possible generalizations and modifications and comments.

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Did you mean Ostrowski's theorem $$|D|>\prod_j \left(|a_{jj}|-\sum_{i\neq j}|a_{ij}|\right)$$ for diagonally dominant matrices (see e.g. http://planetmath.org/propertiesofdiagonallydominantmatrix)?

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  • $\begingroup$ Yes, it is exactly what I wanted, user35593. Thank you very much! $\endgroup$
    – Sergei
    Commented Apr 6, 2016 at 18:14
  • $\begingroup$ Do you know a reference to Ostrowski's paper? Is not a first proposed inequality a consequence? $\endgroup$
    – Sergei
    Commented Apr 6, 2016 at 18:29
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    $\begingroup$ Perhaps it is Ostrowski, Alexander; Über die determinanten mit überwiegender Hauptdiagonale. Comment. Math. Helv. 10 (1937), no. 1, 69–96, or A. M. Ostrowski, "Sur la determination des borns inferieures pour une class des determinants," Bulletin des Sciences Mathmatiques, vol. 61, no. 2, pp. 19–32, 1937. $\endgroup$
    – Gerry Myerson
    Commented Apr 6, 2016 at 23:31
  • $\begingroup$ @Gerry Myerson - thank you. May somebody help with a paper in Bulletin des Sciences Mathematiques? A first paper-not an easy reading... $\endgroup$
    – Sergei
    Commented Apr 7, 2016 at 7:32
  • $\begingroup$ Is the next generalization valid with two pivoted elements: let $(a_1,a_2,a_3,\dots)$ be a line of a determinant with $a_1+a_2>\sum_{k=3}^n a_k$. Is the Ostrowski inequality still valid with a product of \prod(a_1+a_2-\sum_{k=3}^n a_k)$ -??? $\endgroup$
    – Sergei
    Commented Apr 18, 2016 at 16:46

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