I need to show an inequality that I heavily suspect to involve Taylors's formula, but I am failing at proving it.
We are given a function $S(x,\theta)$, where $x\in R^{d_1}$ and $\theta\in\Theta\subset R^{d_0}$, $\Theta$ being bounded, that is symmetric and invertible and its first and second derivatives in $\theta$ exit. We define the function $$Q(x,\theta^*,\theta)=Tr(S(x,\theta^*)^{-1}S(x,\theta)-I_d)-\log\det(S(x,\theta^*)^{-1}S(x,\theta)).$$
I have proven that this function is nonnegative and that its Hessian matrix at $\theta=\theta^*$, i.e. we derive by $\theta$, is nonnegative definite. This is due to the fact that $Q(x,\theta^*,\theta^*)=0$ and therefore the function attains a local minimum at $(x,\theta^*,\theta^*)$ and the Hessian matrix will be nonnegative. We denote this matrix by $\Xi(x,\theta^*)$.
No, I whish to show that there exists a constant $c$ such that
$$Tr(S(x,\theta_0)^{-1}S(x,\theta_1)-I_d)-\log\det(S(x,\theta_0)^{-1}S(x,\theta_1)) \geq (\theta_1-\theta_0)^\top\Xi(x,\theta^*)(\theta_1-\theta_0)+r_n(\theta_1-\theta_0)^\top(\theta_1-\theta_0)$$
where $|r_n|\leq c\vartheta_n$ and $\vartheta_n:=|\theta_1-\theta_0|:=((\theta_1-\theta_0)^\top(\theta_1-\theta_0))^{1/2}$ and $|\theta_k-\theta^*|\leq \vartheta_n, ~k=0,1.$
I have tried all different kinds of things: I tried writing $Q(x,\theta_0,\theta_1)=Q(x,\theta^*+(\theta_0-\theta^*),\theta^*+(\theta_0-\theta^*))$ and applying Taylor's formula up to the second derivative in the point $(\theta^*,\theta^*)$. Here, I encounter two problems: I obtain something like $$Q(x,\theta_0,\theta_1) \geq \frac{1}{2} (\theta_0-\theta^*,\theta_1-\theta^*)^\top \Xi(x,\theta^*) (\theta_0-\theta^*,\theta_1-\theta^*).$$ Firstly, there is this $1/2$ that should not be there and secondly, $(\theta_0-\theta^*,\theta_1-\theta^*)$ is not what we want either. Additionally, Taylor's formula gives us a remainder term of the form $||(\theta_0-\theta^*,\theta_1-\theta^*)||^3$ that has not helped me to obtain the term $r_n$.
I have also tried applying Taylor's formula one by one for $\theta_0$ and $\theta_1$, but not seen anything helpful there either.
I feel like I am missing a fact about Taylor's formula or some kind of special version. Attempts at googling or looking up in books have not helped me thus far.
So, how does one attain such an inequality or are there any forms of Taylor's formula that could help me with this?
I am greatful for any hints or help.