Let $V_1, V_2$ be fixed, real, symmetric and positive definite $p\times p$ matrices. I want to characterize the set of $\mathbb R_+-$valued functions $\gamma_1(t)$ and $\gamma_2(t)$ such that, upon letting $\Omega(t) = \gamma_1(t)V_1 + \gamma_2(t)V_2$:
$$ 2\Omega'(t)\Omega(t)^{-1}\Omega'(t) - \Omega''(t) \geq 0, \forall t\in [0, 1], $$
subject to $\gamma_i(0) = c_i$ and $\gamma_i(1) = k_i$, for some fixed constants $c_i, k_i$. Here, $A \geq 0$ means that $A$ is positive semi-definite and $A'(t): = dA(t)/dt$.
An obvious choice that works for any $V_1, V_2$ is $\gamma_i(t) = c_i(1 - t) + k_it$, i.e. a straight line between the two endpoints. But is there a general solution to the problem that characterizes all possibilities? Solutions are allowed to depend on $V_1, V_2$.