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

Abstract

ABSTRACT. We present results on the relationship between the growth of the maximum modulus and the decay of Taylor coefficients of entire functions of several variables. The results are obtained by two different methods, the first of which had been proposed earlier by Oskolkov for the one-dimensionai case, and the second is based on the use of the Legendre-Jung-Fenchel conjugates of the weight functions. Attention is mainly paid to the characterization of the growth of entire functions with respect to the conjunction of variables; however, some results are obtained for the case in which there is different growth with respect to different variables. 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).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