Embedding theorems and extremal problems for holomorphic functions on circular domains of Cn

The paper concerns complex-valued functions which are holomorphic in bounded complete n-circular domains <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0004.gif"></inline-graphic> and fulfil some geometric conditions. The families <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0005.gif"></inline-graphic> of such kind of functions were considered for instance by Bavrin [1,2], Dobrowolska and Liczberski [4], Dziubiski and Sitarski [5], Fukui [6], Higuchi [8], Jakubowski and Kamiski [9], Liczberski and Wrzesie [14], Marchlewska [15,16], Michiwaki [17], and Stankiewicz [22]. The above functions were applied later to research some families of locally biholomorphic mappings in <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0006.gif"></inline-graphic> (see for instance Pfaltzgraff and Suffridge [19], Liczberski [12], Hamada, Honda and Kohr [7]). In this paper, we consider an interesting family <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0007.gif"></inline-graphic> of the type <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0008.gif"></inline-graphic> which separates two Bavrin's families <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0009.gif"></inline-graphic>. These families correspond to the well-known families of convex univalent and close-to-convex univalent functions of one variable, respectively. We define <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0010.gif"></inline-graphic> using the property of evenness of functions. We obtain for <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0011.gif"></inline-graphic> some embedding theorems relevant to the mentioned separation question. Applying the Minkowski function of <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0012.gif"></inline-graphic>, we solve also some extremal problems for functions from <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0013.gif"></inline-graphic>. As an application, we give a topologic property of the family <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_794139_ilm0014.gif"></inline-graphic>.