Is it possible to show that exp(A)-exp(B) is negative definite provided $A-B$ is negative definite?
The example given in this answer serves as a counterexample. In particular: if we take $$ A = \pmatrix{1 & 0\\0 & 2}, \quad B = \pmatrix{2 + \epsilon & 1\\1 & 3} $$ (e.g. with $\epsilon = 0.01$), we find that $A - B$ is negative definite, but $\exp(A) - \exp(B)$ fails to be negative definite.