Generalized matrix version of reverse Hölder inequality
A linear map \(\phi: {\mathbb C}^{n \times n} \longrightarrow {\mathbb C}^{m \times m}\) is called positive if it maps positive definite matrices (PDM) into PDM. For \(n\)-square matrices \(A\) and \(B\), \(A \geq B\) means that \(A - B\) is positive semidefinite. Furthermore, a binary operation \(\sigma\) defined on the set of PDM is called a connection if (i) \(A \leq C, \; B \leq D\) implies \(A \sigma B \leq C \sigma D,\) (ii) \(C (A \sigma B) C \leq (CAC) \sigma (CBC),\) (iii) \(A_k \searrow A\) and \(B_k \searrow B\) imply \(A_k \sigma B_k \searrow A \sigma B.\) A mean is a connection with the additional condition (iv) \(I \sigma I = I.\) The operator means corresponding to operator monotone functions \(t^{\alpha}, \, 0 < \alpha < 1,\) are called weighted geometric means, denoted by \(\#_{\alpha}.\) The authors present the following results: Let \(A, B > 0\) be such that \(aA \geq B \geq bA\) for some scalars \(0 < b \leq a,\) and let \(\phi\) be a positive linear map. Then for any connection \(\sigma\) we have the following: \[ \phi(A) \sigma \phi(B) \leq \tfrac{1}{\omega} \, \phi(A \sigma B),\tag{1} \] where \(\omega = \frac{f(a) - f(b)}{(a-b)f'(c)}\) for some fixed \(c \in (b,a)\). \(f\) is the representing function of \(\sigma\). \[ \phi(A) \sigma \phi(B) - \phi(A \sigma B) \leq -g(t_0) \phi(A),\tag{2} \] where \(g(t) = \mu t + \nu - f(t),\) \(t_0 \in (b,a)\) a fixed point, \(\mu = \frac{f(a) - f(b)}{a-b}, \; \nu = \frac{af(b) - bf(a)}{a-b}.\) \(f\) has the same meaning as above. By taking \(\sigma = \#_{\alpha}\), (1) implies the main results proved by \textit{E.-Y. Lee} [Linear Algebra Appl. 430, 805--810 (2009; Zbl 1157.15024)], \textit{J.-C. Bourin, E.-Y. Lee, M. Fujii} and \textit{Y. Seo} [Linear Algebra Appl. 431, 2154--2159 (2009; Zbl 1179.15021)], and \textit{M. Niezgoda} [Linear Algebra Appl. 433, 136--142 (2010; Zbl 1193.15018)], and (2) implies the result proved by \textit{M. Fujii, E.-Y. Lee} and \textit{Y. Seo} [Linear Algebra Appl. 432, 2565--2571 (2010; Zbl 1189.15028)].
Hölder/ Cauchy-Schwarz reverse inequality
scientific article; zbMATH DE number 5833971
scientific article