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Someone mentioned the following fact to me today: if $\ln f(x)$ is concave, then the roots of $f(x)$ and $f'(x)$ interlace. The proof is very simple: the assumption implies that $(d/dx)(f'(x)/f(x)) \leq 0$ which in turn implies that $\arg(f(x)+if'(x))$ is always decreasing. Thus $f(x)+if'(x)$ always moves counterclockwise in the complex plane, which implies that the the roots of $f(x)$ and$f'(x)$ interlace.

Is there a way to visualize this? This contains within it the question of how I should visualize log-concavity, which I'm not sure about. So I'm asking for some sort of geometric interpretation of log-concavity which will allow me to also see the interlacing of roots.

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Huh? If $\ln f(x)$ exists (everywhere) at all, then $f(x)$ can have no roots. What does it mean for those non-existent roots to "interlace"? –  Henning Makholm Jan 23 '12 at 4:23

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Henning has already pointed out in a comment that the question doesn't make sense when taken literally. I'll assume that you mean that $\ln |f|$ (note the absolute value) is concave wherever it is defined.

You seem to be assuming that $f$ is differentiable. (In fact the tags seem to indicate that $f$ is supposed to be a polynomial, but this isn't mentioned in the question, so I won't assume it.)

Now if $f$ is differentiable, then quite independent of the behaviour of $\ln |f|$, there has to be an extremum of $f$, and hence a root of $f'$, between any two roots of $f$. Thus there can't be more than one root between two stationary points, and the question is only whether there can be more than one stationary point between two roots.

Indeed your argument shows that this can't happen when $\ln |f|$ is concave, and this is a special case of the fact that log-concavity of a positive differentiable function implies unimodality, and this is a useful geometric interpretation of log-concavity to keep in mind. You can see this more directly by noting that where $f$ has an extremum, $\ln |f|$ has an extremum of the corresponding kind, and since a concave function can't have minima, a positive differentiable log-concave function can't have minima, and thus it can have at most one maximum.

For some applications of this connection in the case of sequences, see generatingfunctionology by Herbert Wilf.

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