The inequality of Milne and its converse. III (Q2316557)

The paper generalizes Milne inequalities that were previously obtained in, for example, [\textit{C. R. Rao}, Linear Algebra Appl. 321, No. 1--3, 307--320 (2000; Zbl 0977.15007)] and also considered in their previous papers [J. Inequal. Appl. 7, No. 4, 603--611 (2002; Zbl 1004.26011); ibid. 2006, Article 21572, 7 p. (2006; Zbl 1092.26010)]. If \(p=(p_1,\dots,p_n)\) are real numbers in \((-1,1)\) and \(w=(w_1,\dots,w_n)\) positive real numbers summing to 1, then it holds for the functions defined by \[ S_w(f(p))=\sum_{j=1}^n w_j f(p_j)\text{ and } T_{w}(p,r,\gamma)=\frac{S_w(\frac{p}{1-p^2})^2}{[\frac{1}{2}S_w(\frac{1}{1-p})^r+\frac{1}{2}S_w(\frac{1}{1+p})^r]^{\gamma/r}},\text{ with }r>1, \] that \[ S_w\left(\frac{1}{1-p^2}\right) -\frac{1}{2}T_{w}(p,r,\alpha) \le \left[S_w\left(\frac{1}{1-p}\right)S_w\left(\frac{1}{1+p}\right)\right]^{1/2} \le S_w\left(\frac{1}{1-p^2}\right) -\frac{1}{2}T_w(p,r,\beta) \] if and only if \(\alpha\le \frac{3}{2(r+1)}\) and \(\beta\ge1\). In particular when \(r=1\), then the \(p_j\) can be replaced by commuting Hermitian matrices \(P_j\) that satisfy \(-I\le P_j\le I\) in the sense of positive definite matrices with \(I\) the identity.