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The desired inequality follows from an majorisation argument : According to Bhatia Matrix Analysis Corollary III.4.6 we have : $log \lambda(AB) \succ log \lambda^\downarrow(A) + log \lambda^\uparro …

No, choose $\Omega=\{z \in \mathbb{C} : |z| \leq 1 \} ,\ f(z)=Re\ z^{n}$ and let $n\rightarrow \infty$ .

Follows from Hölder's inequality (p=6, q = 6/5): $ab^5 + ba^5 \le (a^6+b^6)^{1/6} (b^6+a^6)^{5/6}$

The conjecture is true. Lemma 1 : For every matrix $Z$ holds $\lambda_j(Z^* Z + Z^* + Z ) \leq \lambda_j(Z^* Z + 2 (Z^* Z)^{1/2})$ . For a proof see the proof of $\lambda_j(Z^* + Z ) \leq \lambda_j …

Fedor's proof can be generalized : Let $S = X^{1/2}$, $U_i = V_i S^{1/2}$ and $W_i = U_i (U_i^T S U_i)^{-1/2}$ . Then $W_i^T S W_i = I_m$ where $I_m$ is the $m\times m$ identity . Since $log(x) \ge …

Now my goal is to apply equation (5.21) in Ando, Majorizations and Inequalities in Matrix Theory, http://ac.els-cdn.com/0024379594903417/1-s2.0-0024379594903417-main.pdf? …