Extremal Problem On The Third Cartan-the Hartogs Domain

In the theory of one complex variable, Riemann mapping theorem has resolved the problem about the classification of simply connected domains in the complex plane. While the key step of proving the Riemann mapping theorem is to find the extremal function of the extremal problemwhere Ω is a simply connected domain in C_∞ with at least two boundary points and F|^Ω is the family of all holomorphic functions f such that f maps Ω conformally onto the unit disk.The similar extremal problem is still significant to study in the theory of several complex variables. In this paper we study the extremal problem between the Cartan-Hartogs domain of the third type and the unit hyperball. We obtain the Caratheodory extremal mappings, the Caratheodory extremal and the extremal distances from the Cartan-Hartogs domain of the third type to the unit hyperball. Here the Cartan-Hartogs domain of the third type is following:where (?)_III(q) denotes the third type of Cartan domain in the sense of L.K.Hua, Z is a skew-symmetrical matrix of order q, Z^t denotes the transposed of Z, Z|- denotes the conjugate of Z, det denotes the determinant of a square matrix, N is a positive integer, and K is a positive real number.We firstly obtain the general form of the minimal circumscribed and the maximal inscribed Hermitian ellipsoid of Y_III(N, q, K):where M=(q(q-1))/2Then according to the concrete computing,we obtain the following conclusions: (1)When 0 < K ≤ 2[q/2], we obtain the Caratheodory extremal mapping, the Caratheodory extremal value and when 2 ≤ K ≤ 2[q/2] the Caratheodory extremal distance from Y_III(N, q, K) to the unit hyperball.(2)When K > 2[q/2], we obtain the Caratheodory extremal mapping, Caratheodory extremal value and the Caratheodory extremal distance from Y_III(N,q, K) to the unit hyperball.