Boundary values and zeros of Harmonic Product of Complex-valued Harmonic Functions in a simply connected bounded Domain
Ayantu Guteta Fite, Hunduma Legesse Geleta
TL;DR
This work investigates when the product of two complex-valued harmonic functions remains harmonic in a bounded simply connected domain. It shows that such a product enforces a rigid structure: if $f=f_1+\overline{f_2}$, then the other factor must satisfy $F=\alpha f_1-\alpha \overline{f_2}$ with $\alpha\neq0$, yielding $fF=\alpha f_1^2-\alpha \overline{f_2}^2$ and a sign-reversal relation between their dilatations. The authors derive boundary-value relations, demonstrating that the boundary of $F$ is obtained by swapping and scaling the boundary data of $f$, and that the boundary of the product is a constant multiple of $i$ times the sum of squares of the boundary components. As an auxiliary result, they obtain a sharp bound on the number of zeros for the product when all functions are complex-valued harmonic polynomials, showing at most $2n^2$ zeros for degree $n$ of $f$, which is half of the general Wilmshurst bound. These findings contribute to understanding multipliers and invariant subspaces within the space of complex-valued harmonic functions and have implications for boundary behavior and zero distribution in this setting.
Abstract
The product of two complex-valued harmonic function is not in general complex-valued harmonic function. In this paper we show that if a complex-valued harmonic function is the product of two complex-valued harmonic functions, then it is the difference of two squares, one is analytic and the other is co-analytic. As a result of this, if one of the factors of the product is known, then the other factor is expressed in terms of the known factor explicitly. As an application of this we determine the boundary value of one of the factors and the product provided the boundary value of the other factor is known. It is shown that the boundary value of the product is a pure imaginary constant. Moreover, if such a product and factors are complex-valued harmonic polynomials, then the number of zeros of the product is at most half of the square of its degree which is by half smaller than what is known in the literature about the maximum number of zeros of complex-valued harmonic polynomials. The feature of this paper is to explore multipliers for some subspace of complex-valued harmonic functions and determine some nontrivial invariant subspace.