On Some Questions in the Theory of Entire Functions

A relationship between the maximum modulus and Taylor coefficients of entire functions of several complex variables

“…In recent years, some very general results on the construction of the scale of comparison of entire functions were obtained by Oskolkov [2,3]. In particular, he showed that in some class I of real-valued functions defined on [1, c~), one can introduce the a-type of an entire function and express it in terms of the Taylor coefficients f,.…”

“…It is the characterization of growth of entire functions in C '~ , rn > 1, that is the central topic of this paper. First, on the basis of the methods developed in [2,3] for the onedimensional case, analogs of one-dimensional results for the growth of entire functions in C m with respect to the conjunction of variables are obtained. It should be noted that some of the results in this direction are new even for m ----1.…”

“…In connection with [2,3], the question arose as to whether similar results are valid for other function series: one-dimensional and multidimensional Dirichlet or Taylor-Dirichlet series, etc. Perhaps, in trying…”

“…and express it in terms of the Taylor coefficients f,. Furthermore, for an arbitrary entire function f, there exists a function a0 of class I such that 0 < ~ra0(f) < q-co [2,3]. Oskolkov also showed that for any entire function f the required weight function a0 (with ~0 (f) = 1) can be chosen in a class I narrower than the original class I (the exact description of the classes 7 and I introduced in [2,3] is given below).…”

“…Furthermore, for an arbitrary entire function f, there exists a function a0 of class I such that 0 < ~ra0(f) < q-co [2,3]. Oskolkov also showed that for any entire function f the required weight function a0 (with ~0 (f) = 1) can be chosen in a class I narrower than the original class I (the exact description of the classes 7 and I introduced in [2,3] is given below).In connection with [2,3], the question arose as to whether similar results are valid for other function series: one-dimensional and multidimensional Dirichlet or Taylor-Dirichlet series, etc. Perhaps, in trying…”