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

Abtin Daghighi

• 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.