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

• Autores: Abtin Daghighi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 69, Fasc. 2, 2018, págs. 173-180
• Idioma: inglés
• DOI: 10.1007/s13348-017-0197-3
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• Resumen

  • Let ?⊂ℂ be a domain and let ?(?)=?(?)+?¯?(?), where a, b are holomorphic for ?∈?. Denote by ? the set of points in ? at which |?| attains weak local maximum and denote by ? the set of points in ? at which |?| attains strict local maximum. We prove that for each ?∈?∖? , |?(?)|=∣∣∣(∂?∂?+?¯∂?∂?)(?)∣∣∣ Furthermore, if there is a real analytic curve ?:?→?∖? (where I is an open real interval), if a, b are complex polynomials, and if ?∘? has a complex polynomial extension, then either f is constant or ? has constant curvature.