Let $f$ be a real analytic function on $|x|< R$.
Let $N(R,f)$ be the number of zeros of $f$ in the region of analyticity.
Can we show taht
\begin{align}
N(R,f) \le N(R,f^\prime)+2
\end{align}
wher $f^\prime$ is a derivative of $f$.
Because $f$ is analytic we know that the number of zeros $N(R,f)$ and the number of critical points $N(R,f)$ is finite (this follows from Liouville's theorem).
How to cleanly argue the above bound? Also, a reference would be appreciated.