Let $V : (a,b) to mathbb{R}$ be smooth, strictly increasing and $V(a) = 0$. Suppose that $f : (a,b) to mathbb{R}$ is smooth and satisfies $f^{prime prime} (x) + V(x) f(x) = 0$ on $(a,b)$. Can we then bound $sup_{x in (a,b)} |f(x)|$ in terms of $f(a) , f(b) , f^{prime} (b)$? I intentionally don’t put $f^{prime} (a)$ into the list.