Geodesic
Sketch of the theory of growth of functions holomorphic in a multidimensional torus
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex analysis, Tome 142 (2017), pp. 88-101.

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We develop an approach to the theory of growth of class-$H(\mathbb{T}^n)$ functions holomorphic in a multidimensional torus $\mathbb{T}^n$ based on the structure of elements of this class and well-known results of the theory of growth of entire functions of several complex variables. This approach is illustrated in the case where the growth of the function $g\in H(\mathbb{T}^n)$ is compared with the growth of its maximum modulus on the skeleton of polydisk. Properties of the corresponding characteristics of growth of class-$H(\mathbb {T}^n)$ functions are examined and their relation to coefficients of their Laurent series are studied. A comparative analysis of these results and similar assertions of the theory of growth of entire functions of several variables is given.
Keywords: entire function of several variables, holomorphic function in multidimensional torus, convex function, characteristics of growth, carrier, strictly convex cone.
Mots-clés : multiple Laurent series 
@article{INTO_2017_142_a7,
     author = {M. N. Zav'yalov and L.S. Maergoiz},
     title = {Sketch of the theory of growth of functions holomorphic in a multidimensional torus},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {88--101},
     publisher = {mathdoc},
     volume = {142},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2017_142_a7/}
}
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M. N. Zav'yalov; L.S. Maergoiz. Sketch of the theory of growth of functions holomorphic in a multidimensional torus. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Complex analysis, Tome 142 (2017), pp. 88-101. http://geodesic.mathdoc.fr/item/INTO_2017_142_a7/