All functions are smooth and continuous.
I use ' to signfy $d/dt$.
Given:
q_i'(0)=0, for i=1,2,3,4.
$q_2(t)q_3(t)>q_1(t)q_4(t)$ for $0<=t
q_4(0)*q_3''(0)>q_4''(0)*q_3(0)
(q_4''(0)+q_2''(0))(q_1(0)+q_3(0)>(q_4(0)+q_2(0))(q_1''(0)+q_3''(0))
This also holds but I suspect it is superfluous:
q_2(0)*q_1''(0)
One can also add this without loss of generality:
$q_1(0)=1, q_2(0)=-1$
Prove or disprove that:
q_4(t)*q_3''(t)>q_4''(t)*q_3(t)
(q_4''(t)+q_2''(t))(q_1(t)+q_3(t)>(q_4(t)+q_2(t))(q_1''(t)+q_3''(t))
for $0<=t
Counterexample will do.