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 \(p\in {\varLambda }\setminus {\varSigma }\),
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.
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Notes
The terminology n-analytic (as opposed to polyanalytic of order n) can be found in early important work in this field by e.g. Bosch and Kraikiewvicz [4] and more recent work by e.g. Ramazanov [12, 13] and Cuckovic and Le [6]. These authors seem to agree that the term polyanalytic is useful to describe the general case when the order is not of particular interest. Also one advantage is the possibility of a short and concise generalization to several complex variables given by \(\alpha \)-analytic functions (where \(\alpha \) is allowed to be a multi-index) instead of having to explaining what the order is with respect to each variable separately.
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Daghighi, A. A necessary condition for weak maximum modulus sets of 2-analytic functions. Collect. Math. 69, 173–180 (2018). https://doi.org/10.1007/s13348-017-0197-3
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DOI: https://doi.org/10.1007/s13348-017-0197-3