Maximum Modulus Points of Entire Functions: Angular Oscillation and Circle Reflection

Abstract

Let f be a non-monomial entire function and let v_f(r) denote the number of points on |z| = r where |f| attains its maximum. We study the finite-lower-order form of the Erdős maximum-modulus problem. For 0 < η < 1, an elementary variation argument gives an M(r,f)-relative bound for the number of relevant near-maximum components on zero-free circles. We then show that a large cluster of exact maximum-modulus points forces many angular critical points of log |f|, represented by the logarithmic derivative zf′(z)/f(z). A conjugate circle-reflection operator, valid for entire functions with arbitrary complex coefficients, converts these angular critical points into zeros of an explicitly defined holomorphic function on C \ {0}. For the class P(z)e^{Q(z)}, we prove a uniform bound for the number of maximum-modulus points at regular radii. We also record a restricted genus-0 variation estimate under explicit auxiliary hypotheses, retaining its logarithmic loss. The general finite-lower-order variation estimate needed for the full problem remains open; accordingly, no solution of the Erdős problem is claimed.