I have a question when I try to have some analysis on the following expression. \begin{align} \mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\frac{1}{N_0+\sigma^2}\mathbf{X}\right)\right], \end{align} where $\mathbb{E}$ denotes the expectation, $\log_2$ denotes the log function based on 2, $\det (\cdot)$ denotes the determinant operation, $\mathbf{I}_L$ is a identity matrix of size $L\times L$, $\mathbf{X}$ is a non-diagonal complex matrix of size $L\times L$ follows Wishard distribution, $\sigma^2=\frac{1}{L}{\rm{Tr}(\mathbf{Y})}$, $\mathbf{Y}$ is a non-diagonal complex matrix of size $L\times L$ follows Wishard distribution, $\mathbf{X}$ and $\mathbf{Y}$ are independent, ${\rm{Tr (\cdot)}}$ denotes trace operation, and $N_0$ is just a real constant.
My question is can I have the following inequality based on Jensen's inequality? OR ANY UPPPER/LOWER BOUND ON THE ABOVE EXPRESSION? \begin{align} \mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\frac{1}{N_0+\sigma^2}\mathbf{X}\right)\right]\le\log_2\det\left(\mathbf{I}_L+\frac{1}{N_0+\mathbb{E}\left[\sigma^2\right]}\mathbb{E}\left[\mathbf{X}\right]\right) \end{align}
For context, this is related to the signal-to-interference-pulse-noise-ratio (SINR) in wireless communications.