Collectanea Mathematica

, Volume 69, Issue 2, pp 173–180

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

• Abtin Daghighi
Article

Abstract

Let $${\varOmega }\subset \mathbb {C}$$ be a domain and let $$f(z)=a(z)+\bar{z}b(z),$$ where ab 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 ab are complex polynomials, and if $$f\circ \kappa$$ has a complex polynomial extension, then either f is constant or $$\kappa$$ has constant curvature.

Keywords

Polyanalytic functions q-Analytic functions Peak sets Maximum modulus sets

Mathematics Subject Classification

30G30 35B50 35G05

Notes

Acknowledgements

The author is grateful to the referee for helpful comments and new insights.

Conflict of interest

The author declares no conflict of interest.

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