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For this problem, suppose $g:\mathbb{R}\rightarrow\mathbb{R}$ is such that $g\in\mathcal{C}^{k}(\mathbb{R})$, and there exists $\epsilon>0$ such that

\begin{align*} \epsilon<|g^{(k)}(x)|<\epsilon^{-1}, \ \forall x\in\mathbb{R}.\end{align*}

Notice in the case $k=1$, $g$ necessarily has a unique root. Call this root $z_{1}$. By the Fundamental Theorem of Calculus, we may write

\begin{align*}g(x)=(x-z_{1})\int_{0}^{1}g'(z_{1}+(x-z_{1})t)dt.\end{align*}Using our regularity condition on $g'$, we may conclude that

\begin{align}C|x-z_{1}|\le |g(x)|\le C^{-1}|x-z_{1}|,\end{align}where $C$ depends only on $\epsilon$.

In the case general case $k\ge 2$, we are no longer guaranteed $g$ has a specific number of roots, merely that the number of roots does not exceed $k$. Nonetheless, if there are any, we may play a similar game with the FTC: let $z_{1},...,z_{N}$ denote the roots, where $N\le k$. Then generalizing the equation above,we can show

\begin{align*}g(x)=\prod_{j=1}^{N}(x-z_{j})\int_{[0,1]^{N}}g^{(N)}(z_{1}+(z_{2}-z_{1})t_{1}+(z_{3}-z_{2})t_{1}t_{2}+...+(z_{N}-z_{N-1})t_{1}...t_{N-1}+(x-z_{N})t_{1}...t_{N})t_{1}^{N-1}t_{2}^{N-2}...t_{N-1}dt_{1}...dt_{N}.\end{align*}

If $k=N$, we can use the regularity on $g^{(k)}$ to get a bound on $g$ similar to the one above for $k=1$. This is very convenient, since it gives us concrete bounds on $\frac{1}{g}$, away from the roots of $g$. In particular, the bound will be of the form

\begin{align*}C\prod_{j=1}^{k}|x-z_{j}|^{-1}\le\left|\frac{1}{g(x)}\right|\le C^{-1}\prod_{j=1}^{k}|x-z_{j}|^{-1}\end{align*}where $C$ depends only on $\epsilon$. Thus, in the case of $g$ having "maximal" roots for the given regularity, we get a precise bound on the behavior of $\frac{1}{g}$.

In the case of fewer than $k$ roots, things are less clear to me. My question is: are there any known results or techniques that could allow me to derive bounds of this form for the case of fewer than $k$ roots? Taylor's theorem gives us some foothold, by telling us $g$ is bounded by polynomials of order $k$ whose leading coefficients depend only on $\epsilon$. However, these polynomials change from point-to-point, and require knowledge of lower-order derivatives of $g$. As such, these bounds are not as strong as the ones we get from the FTC trick in the case of maximal roots. Something tells me there should be be similar bounds in this case of fewer than $k$ roots, where the points $z_{j}$ are replaced by points where higher order derivatives vanish. I have as yet been unable to show this. Any thoughts?

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