Let $R$ be any quantum state and $X$ any observable, then we define $$F(R, X) = 4 \operatorname{Tr} \left( L^2 R \right)$$ as the quantum Fisher information, where $L$ is the logarithmic derivative determined by $$i(RX - XR) = \frac{LR + RL}{2}$$
I know that the Fisher information is a convex function but I can not prove this. Please help me.
Typically the quantum Fisher information is defined as $$ F(\rho(x)) = \operatorname{Tr}(\rho(x) L^2), $$ where $x\in \mathbb R$ is an unknown parameter and where $L$ is the symmetric logarithmic derivative, which is implicitly defined as $$ \frac{d}{dx} \rho = \frac12(\rho L + L \rho). $$ For your problem, $L$ is not really of concern. That the Fisher information is convex in $\rho$ is a simple consequence of the linearity of the trace. And all linear functions are convex.