A necessary condition for weak maximum modulus sets of 2-analytic functions

Abstract

Let ${\varOmega }\subset \mathbb {C}$ be a domain and let $f(z)=a(z)+\bar{z}b(z),$ where a, b are holomorphic for $z\in {\varOmega }.$ Denote by ${\varLambda }$ the set of points in ${\varOmega }$ at which $\left| f\right|$ attains weak local maximum and denote by ${\varSigma }$ the set of points in ${\varOmega }$ at which $\left| f\right|$ attains strict local maximum. We prove that for each $p\in {\varLambda }\setminus {\varSigma }$,

$$\begin{aligned} \left| b(p)\right| =\left| \left( \frac{\partial a}{\partial z} +\bar{z}\frac{\partial b}{\partial z}\right) (p)\right| \end{aligned}$$

Furthermore, if there is a real analytic curve $\kappa :I\rightarrow {\varLambda }\setminus {\varSigma }$ (where I is an open real interval), if a, b are complex polynomials, and if $f\circ \kappa$ has a complex polynomial extension, then either f is constant or $\kappa$ has constant curvature.