A converse of a matrix inequality (Q5929766)

Let \(X = (x_{ij})\) denote a nonnegative \(m \times n\)-matrix whose entries do not all vanish. Let \(s(X)\) denote the sum of all entries of \(X\) and let \(\|X \|\) be the Frobenius norm of \(X.\) Then the following inequality holds: NEWLINE\[NEWLINE \frac{m \sum_{i=1}^m \left( \sum_{j=1}^n x_{ij} \right)^2 + n \sum_{j=1}^n \left( \sum_{i=1}^m x_{ij} \right)^2}{(s(X))^2 + mn \|X \|^2} \geq \frac{m+n}{mn + \text{ min}(m,n)}. \tag \(*\) NEWLINE\]NEWLINE This estimate is best possible; conditions for resp. a characterization of the case of equality are given explicitly. That the quotient on the left hand side of \((*)\) admits the upper bound 1 was established earlier by \textit{E. R. van Dam} [Linear Algebra Appl. 280, No. 2-3, 163-172 (1998; Zbl 0934.15021)].