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Complex harmonic mean

Atsushi Nakayasu

TL;DR

This work defines the complex harmonic mean $\mathbf{H}[Z] = \mathbf{E}[Z^{-1}]^{-1}$ for nonzero complex random variables and shows that, unlike the positive-valued case, $\mathbf{H}[Z]$ can lie outside the convex hull of the range. It establishes strong geometric bounds via inversion: if $\operatorname{Range}[Z]$ lies in a disk not containing $0$, then $\mathbf{H}[Z]$ remains in that disk. The paper also derives sharp modulus and real-part estimates, namely $|\mathbf{H}[Z]| \ge \mathbf{H}[|Z|]$ and $\Re\mathbf{H}[Z] \ge \mathbf{H}[\Re Z]$, with equality in precise scalar cases. In the two-point setting, $\mathbf{H}[Z]$ traces a circle through the two support points and $0$, or a line segment when the points are collinear with $0$, with explicit examples illustrating the geometry. These results have potential applications in homogenization of complex-valued equations and provide a geometric lens for understanding complex effective coefficients.

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.

Complex harmonic mean

TL;DR

This work defines the complex harmonic mean for nonzero complex random variables and shows that, unlike the positive-valued case, can lie outside the convex hull of the range. It establishes strong geometric bounds via inversion: if lies in a disk not containing , then remains in that disk. The paper also derives sharp modulus and real-part estimates, namely and , with equality in precise scalar cases. In the two-point setting, traces a circle through the two support points and , or a line segment when the points are collinear with , with explicit examples illustrating the geometry. These results have potential applications in homogenization of complex-valued equations and provide a geometric lens for understanding complex effective coefficients.

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.
Paper Structure (6 sections, 4 theorems, 34 equations, 2 figures)

This paper contains 6 sections, 4 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Examples
  3. Estimates by modulus
  4. Estimates by real part
  5. Bounds by circles
  6. Two-point cases revisited

Key Result

Theorem 3.1

For a non-zero complex-valued random variable $Z$, which has a complex harmonic mean $\mathbf{H}[Z]$, the inequality e_estabs holds. The equality holds when $Z = v X$ for some (constant) non-zero complex number $v$ and positive-valued random variable $X$.

Figures (2)

  • Figure 1: Complex harmonic mean of $c_1 = 1+i$ and $c_2 = 1-i$ with the weight $\theta = 0.0, 0.1, \cdots, 1.0$.
  • Figure 2: Complex harmonic mean of $c_1 = 8$ and $c_2 = 1+i$ with the weight $\theta = 0.0, 0.1, \cdots, 1.0$.

Theorems & Definitions (11)

  • Example 2.1
  • Example 2.2
  • Example 2.3: Complex lognormal distribution
  • Theorem 3.1: Estimates by modulus
  • proof
  • Theorem 4.1: Estimates by real part
  • proof
  • Theorem 5.1: Bounds by circles
  • proof
  • Remark 5.2
  • ...and 1 more