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

Let Omega subset of C be a domain and let f (z) = a(z) + (z) over barb(z), where a, b are holomorphic for z is an element of Omega. Denote by. the set of points in Omega at which vertical bar f vertical bar| attains weak local maximum and denote by Sigma the set of points in Omega at which vertical bar f vertical bar attains strict local maximum. We prove that for each p is an element of Lambda \ Sigma, Furthermore, if there is a real analytic curve kappa : I Lambda\ Sigma ( where I is an open real interval), if a, b are complex polynomials, and if f o kappa has a complex polynomial extension, then either f is constant or kappa has constant curvature.