Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is symmetric and $u^TAu\ge 0$ for all $u\in\mathbb R^2$. We consider the distance between $A,B\in \mathcal S_2$ given by $$d(A,B):={\rm Tr}(A + B - 2\sqrt{\sqrt{A}B\sqrt{A}}),$$ where ${\rm Tr}$ denotes the trace operator. My question is as follows:
Set $u_{\theta}:=\big(\cos({\theta}),\sin(\theta)\big)^T$ for ${\theta}\in [0,2\pi]$. Define $$L(A,B):=\int_0^{2\pi} \Big\{ u_{\theta}^TAu_{\theta}+u_{\theta}^TBu_{\theta}-2\sqrt{u_{\theta}^TAu_{\theta} u_{\theta}^TBu_{\theta}}\Big\} d\theta.$$ Does the maximisation problem below admit a finite upper bound? $$\sup_{A, B\in \mathcal S_2}\frac{d(A,B)}{L(A,B)}<\infty?$$
