= Circle geometric mean of a holomorphic function
{title2=$G_f(R)=\exp L_f(R)$}
The continuous angular <geometric mean> of the positive boundary <complex modulus> is
$$
G_f(R)=\exp\left(\frac{1}{2\pi}\int_0^{2\pi}\log|f(Re^{i\theta})|\,d\theta\right)
=|f(0)|\prod_{j=1}^{N}\frac{R}{|a_j|}.
$$
Here the empty product is one. This differs from the angular <arithmetic mean> $A_f(R)=(2\pi)^{-1}\int_0^{2\pi}|f(Re^{i\theta})|\,d\theta$. In general $G_f(R)\le A_f(R)$ by the <concavity> of the <logarithm>; equality requires constant boundary <complex modulus>. For a <holomorphic function> with no <zeros of a function> in the closed <disk>, <Jensen's formula> gives $G_f(R)=|f(0)|$, even when the boundary <complex modulus> varies with angle. Thus replacing the mean of a <logarithm> by the <logarithm> of an <arithmetic mean> loses the exact zero identity.
Back to article page
