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This is a Quantitative Study study.

September 29, 2025Open Access

Boundary values and zeros of Harmonic Product of Complex-valued Harmonic Functions in a simply connected bounded Domain

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This analysis reveals boundary values and zeros in harmonic products of complex-valued functions, highlighting new relationships.

The product of two complex-valued harmonic functions results in a non-harmonic function, and this study explores its boundaries.
The boundary value of the product is shown to be a pure imaginary constant, revealing new insights into harmonic functions.
For complex-valued harmonic polynomials, the maximum number of zeros is at most half the square of its degree, contradicting previous literature.
This research also explores multipliers and invariant subspaces of complex-valued harmonic functions.

Fite et al. (2025) studied this question.

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