The geometric mean is a Bernstein function

In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers. From this integral representation, the geometric mean is proved to be a Bernstein function and a new proof of the well known AG inequality is provided.Comment: 10 page

Anomalous diffusion of optical vortices in random wavefields

We investigate the dynamic behavior of optical vortices, or phase singularities, in random wavefields and demonstrate the direct experimental observation of the anomalous diffusion of optical vortices. The observed subdiffusion of optical vortices show excellent agreement with the fractional Brownian motion, a Gaussian process. Paradoxically, the vortex displacements are observed exhibiting a non-Gaussian heavy-tailed distribution. We also tune the extent of subdiffusion and non-Gaussianity of optical vortex by varying the viscoelasticity of light scattering media. This complex motion of optical vortices is reminiscent of particles in viscoelastic environments suggesting a vortex tracking based microrheology approach. The fractional Brownian yet non-Gaussian subdiffusion of optical vortices may not only offer insights into the dynamics of phase singularities, but also contribute to the understanding certain general physics, including vortex diffusion in fluids and the decoupling between Brownian and Gaussian