Suppose we have two symmetric positive definite matrices $A,B$ (not simultaneously diagonalizable). How can I find a matrix function $f(t), t\in [0,1]$, such that $f(0)=A, f(1)=B$, and the minimum eigenvalue of $\lambda_{min}(f(t))$ is smoothly interpolating the minimum eigenvalues of $A,B$ such that $$\min(\lambda_{min}(A),\lambda_{min}(B)) \leq \lambda_{min}(f(t)) \leq \max(\lambda_{min}(A),\lambda_{min}(B))?$$ If it is possible to find $f(t)$ such that this is interpolation $\lambda_{min}(f(t))$ is concave, that would be even better.

The naive solution to interpolation would be linear, i.e. $f(t) = (1-t)A + tB$, but that does not satisfy that condition above even though it is concave, i.e. the minimum eigenvalue can easily get larger than the $\max(\lambda_{min}(A),\lambda_{min}(B))$.