For a project I am working on, I am able to show that the derivative of a function $f$ is bounded by: $$1 - C_1 f(x) \leq f'(x) \leq 1 - C_2 f(x)$$ for some constants $C_1,C_2$. I also know that $f(0) = 0$. Let $g_1$ be the solution to the ODE $1 - C_1 g_1(x) = g_1'(x)$ with initial condition $g_1(0) = 0$ and let $g_2$ be the solution to the ODE $1 - C_2 g_2(x) = g_2'(x)$ with initial condition $g_2(0) = 0$. Intuitively, I would think that this information suffices (perhaps with some regularity conditions on $f$) to conclude that $g_1(x)\leq f(x)\leq g_2(x)$. However, I am struggling to find a proof of this fact. Is there a result I can appeal to to make this conclusion?
2026-03-28 01:13:35.1774660415
Bounds on solutions to an ordinary differential equation
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