Voorhoeve index

The Voorhoeve index ${\displaystyle V_{I}(f)}$ of a complex-valued function f that is analytic in a complex neighbourhood of the real interval ${\displaystyle I}$ = [a, b] is given by

${\displaystyle V_{I}(f)={\frac {1}{2\pi }}\int _{a}^{b}\!\left|{\frac {d}{dt}}{\rm {Arg}}\,f(t)\right|\,\,dt\,={\frac {1}{2\pi }}\int _{a}^{b}\!\left|{\rm {Im}}\left({\frac {f'}{f}}\right)\right|\,dt.}$

(Different authors use different normalization factors.)

Rolle's theorem states that if ${\displaystyle f}$ is a continuously differentiable real-valued function on the real line, and ${\displaystyle f(a)=}$ ${\displaystyle f(b)=0}$, where ${\displaystyle a<b}$, then its derivative ${\displaystyle f'}$ has a zero strictly between ${\displaystyle a}$ and ${\displaystyle b}$. Or, more generally, if ${\displaystyle N_{I}(f)}$ denotes the number of zeros of the continuously differentiable function ${\displaystyle f}$ on the interval ${\displaystyle I}$, then ${\displaystyle N_{I}(f)\leq N_{I}(f')+1.}$

Now one has the analogue of Rolle's theorem:

${\displaystyle V_{I}(f)\leq V_{I}(f')+{\frac {1}{2}}.}$

This leads to bounds on the number of zeros of an analytic function in a complex region.