Complex harmonic mean

TL;DR

This paper investigates the properties of the harmonic mean for non-zero complex-valued random variables, establishing geometric bounds and exploring how the mean behaves within certain complex regions.

Contribution

It introduces new geometric estimates and bounds for complex harmonic means, extending classical results to the complex domain and analyzing their behavior within specific regions.

Findings

Complex harmonic means can lie outside the convex hull of the range.

If the range is in a disk not containing zero, the harmonic mean stays in that disk.

Explicit analysis of the two-point case reveals a circular structure.

Abstract

We study the harmonic mean of non-zero complex-valued random variables (complex harmonic mean) and establish several geometric estimates and bounds. In contrast to the classical positive-valued case, complex harmonic means may lie outside the convex hull of the range. We prove that if the range is contained in a disk not containing the origin, then the complex harmonic mean is confined to the same disk. This result is based on the behavior of disks under inversion and convexity arguments. Further estimates involving the modulus and the real part are obtained, and the two-point case is analyzed explicitly, revealing a circular structure. Several examples are provided to illustrate the distinctive features of complex harmonic means.