Mathematics & Physics 2015, 8(1), 75–85 УДК 517.55 A Note on a Distance Function in Bergman Type Analytic Function Spaces of Several Variables Romi F. Shamoyan∗ Sergey M.Kurilenko† Laboratory of complex and functional analysis Bryansk State University Bezhitskaya, 14, Bryansk, 241036 Russia Received 10.08.2014, received in revised form 10.10.2014, accepted 20.11.2014 New sharp estimate concerning distance function in certain Bergman-type spaces of analytic functions on tube domains over symmetric cones is obtained. <...> This is the first result of this type for tube domains over symmetric cones. <...> New similar results in analytic mixed norm spaces on products of tube domains over symmetric cones will also be provided. <...> Introduction and preliminaries In this note we obtain a sharp distance estimate in spaces of analytic functions in tube domains over symmetric cones. <...> This line of investigation can be considered as continuation of previous papers (see, for example, [1,2] and [3] and references there). <...> This new results are contained in the second and third section of this note. <...> We remark that for the first time in literature we consider this type extremal problem related with distance estimates in spaces of analytic functions on tube domains over symmetric cones. <...> In one dimensional tubular domain which is upperhalfspace C+ (see, for example, [4]) our theorem is not new and it was obtained recently in [5]. <...> Moreover arguments we provided below in proof are similar to those we have in one dimension and the base of proof is again the so-called Bergman reproducing formula, but in tubular domain over symmetric cone (see, for example, [4] for this integral representation). <...> Recently various papers appeared where arguments which can be seen in [6] were extended in various directions (see, for example, [1–3]). <...> In particular in mentioned papers various new results on distances for analytic function spaces in higher dimension (unit ball and polydisk ) were obtained. <...> All rights reserved c – 75 – Romi F.Shamoyan, Sergey M.Kurilenko A Note on a Distance Function in Bergman Type . <...> Later several new sharp results for harmonic functions of several <...>