A new proof of some matrix inequalities

In this paper we give alternate proofs of some well-known matrix inequalities. In particular, we show that under certain conditions the inequality holds \begin{align}\sum \limits_{\lambda_i\in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(ab^{T})}$\mathrm{m}\mathrm{i}\mathrm{n}${\log |t-\lambda_i|}{[||a||,||b||]}&\leq # $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \&+\frac{1}{||b||-||a||}\sum \limits{\lambda_i\in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}Also under the same condition, the inequality also holds\begin{align}\int \limits_{||a||}^{||b||}\log|$\mathrm{d}\mathrm{e}\mathrm{t}$(ab^{T}-tI)|dt&\leq # $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(ab^T)(||b||-||a||)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \&+\sum \limits_{\lambda_i\in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}