Influence of Coefficient Arguments on the Maximum Modulus of Entire Functions of Several Variables

This paper investigates how the phases, or arguments, of coefficients in multiple power series influence the maximum modulus of entire functions of several complex variables. While classical complex analysis often emphasizes the absolute sizes of these coefficients, this study highlights that altering only the angles of Taylor coefficients can radically accelerate or amplify a function’s overall growth. To explore this phenomenon, we analyze the relationship between the actual maximum modulus of a function and its majorant. The majorant is defined mathematically as some power series constructed from primary series where every coefficient is replaced by its absolute value. This represents the “maximum possible” growth a function when all coefficients are strictly positive and add together at a given point without any mutual cancellation. The study establishes precise necessary and sufficient conditions under which specific asymptotic inequalities comparing these two distinct growth measures hold true. The main result establishes the sufficient conditions required to bound the growth of the majorant relative to the maximum modulus. Complementing this, other result demonstrates the exactness of these boundaries by proving that if the sufficient conditions are not met, an entire function can be explicitly constructed that strictly exceeds those growth constraints. The research generalizes known mathematical theorems from one-variable case into multidimensional contexts. The exact dependence of a function’s asymptotic growth rate on its coefficient phases is detailed and formulated for broad classes of multivariate transcendental entire functions. Furthermore, the paper establishes the relationship between the function’s maximum value and its majorant while systematically excluding specific, negligible intervals of radii where abnormal growth spikes might otherwise complicate the analysis. Such excluded intervals are a set of finite logarithmic measure. These generalized multidimensional estimates provide a base for controlling the behavior of analytic solutions to partial differential equations when their coefficient phases are perturbed.