Entire Functions of Several Variables: Wiman–Valiron-Type Results Without Exceptional Sets

This article is devoted to establishing analogs of the main theorems of the Wiman–Valiron theory on asymptotic relations for entire functions of several complex variables that are satisfied without exceptional sets when the complex space is exhausted by a system of multiple-circular domains or by a system of A-like polylinear domains. These results concern: (i) analogs of the asymptotic equality of the logarithms of the maximum modulus of an entire function on the boundary of multiple-circular domains of exhaustion and the corresponding maximal term; (ii) a statement describing the behavior of the entire function F(z) of several complex variables z=(z1,…,zp) in the vicinity of the point w, where the value of F(w) is close to the supremum of its modulus on the boundary of polylinear domains; (iii) the so-called fundamental relation of the Wiman–Valiron theory, which establishes the asymptotic behavior of directional derivatives for a given fixed direction of the vector A, which specifies the direction of space exhaustion.