Let $X \in S^{n}_{+}$ be a positive semi-definite matrix with $X_{ii} = 1$ for all $i \leq n$ (thus $X$ is a correlation matrix). Since $X$ is positive semi-definite, we have $|X_{ij}| \leq 1$ for any entry $(i,j)$.

Let $\arcsin(X) = (\arcsin(X_{ij}))_{i,j \leq n}$, i.e., the arcsin function applies on the matrix X element-wisely. From the Schur product theorem, we know $\arcsin(X) \succeq X$, where $A \succeq B$ means $A-B$ is positive semi-definite. I wonder, do we have an inequality on the other side, i.e., there exists some real number $a> 0$ such that $a X \succeq \arcsin(X)$?

I am asking this question because SDP relaxation usually provides good approximation algorithms to combinatorial optimization with a maximization objective. I am thinking about if we can still apply SDP to minimization problems.