Let $I=[0,b)$, $b< \infty$. Suppose $u$ is a positive bounded measurable function on $I$. $v(s)$ is a positive, smooth function on $I$. Note that $u(b),v(b)$ may be $0$. Suppose that $$ u(t) \leqslant u(t_0) +\int_{t_0}^t u(s) \frac{v'(s)}{v(s)}ds $$ for any $0<t_0<t <b$. Then does the inequality below holds? $$ \frac{u(s_2)}{u(s_1)} \leqslant \frac{v(s_2)}{v(s_1)} $$ for any $0 \leqslant s_1 <s_2<b$.

If not, what other conditions should we add to $v(s)$, is $v'(s)\leqslant 0$ enough?