Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. Show that, if $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) > 0$, then there exists $\eta > 0$, such that for all $t \in ] - \eta , \eta [$,
$$\frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \geq \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A ) - \operatorname { Tr } \left( A ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A ) t$$