Suppose we have two functions, $f,g:\sf{R}^{n \times n} \to \sf R$, i.e. a mapping between matrices of dimension of $n$ times $n$ and a real number. Here we can assume that the matrices are symmetric and positive definite. Further, $f$ is a linear function of the matrices, and $g$ is a concave function of the matrix. What is the correlation between $f$ and $g$, i.e. $\rho(f,g)$.
For example, $\mathbf{X}$ denotes the matrix. What can we say about $\rho (tr(X), \log|\mathbf{I}+X|)$.
Intuitively think, $\rho (tr(X), \log|\mathbf{I}+X|)$ should be positive, i.e. $1\geq\rho (tr(X), \log|\mathbf{I}+X|) \geq 0$. But I could not prove that.
Thanks in advance.
Since there is no answer available here, I would like to add some points I found.
For a vector $\mathbf{x} \in \mathbb{R}^{n}$ and $f(\mathbf{x}),g(\mathbf{x}): \mathbb{R}^{n} \to \mathbb{R}$, then function $f$ and $g$ are positively correlated, i.e. $\text{cov} (f,g) \geq 0$, when the distribution of $\mathbf{x}$ is multivariate totally positive of order $2$ (MTP2) and both $f$ and $g$ are montonically increasing or reading [1].
Furthermore, it is shown in [1] that many distributions satisfy the MTP2 property, such as eigenvalues of Wishart, Gaussian.
If the expressions, such as $\log |\mathbf{I} + \mathbf{X}|$, can be expressed as a function of the eigenvalues of $\mathbf{X}$, then it will be easy to deal with.
[1]Karlin, Samuel, and Yosef Rinott. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions." Journal of Multivariate Analysis 10.4 (1980): 467-498.