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.


If $$ x_1 < x_2 < \ldots < x_n $$ are the zeros of $f$ in $|x| < R$, with multiplicities $k_1, k_2, \ldots, k_n$, then $f'$ has zeros of multiplicity $k_j-1$ at $x_j$, and also at least one zero in each interval $(x_j, x_{j+1})$, because of Rolle's theorem. This gives $$ N(R, f') \ge (k_1 - 1) + \ldots (k_n-1) + (n-1) = (k_1 + \ldots + k_n) - 1 = N(R, f) -1 \, , $$ i.e. $N(R, f) \le N(R, f') + 1$.


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