On the Lu Qi-Keng conjecture and the Bergman representative domains

Lu Qi-Keng conjectue and Hua domain

“…And in 1998, Pflug and Youssfi [10] proved that the minimal ball{z ∈ C n : |z| 2 + |zz | < 1} is not a Lu Qi-Keng domain if n 4, this negates the fact proved in [16].…”

“…In 1999, Boas, Fu and Straube [12] proved that the domain {z ∈ C 2 : |z 1 | + |z 2 | 2/p < 1} is not a Lu Qi-Keng domain if p > 2, it contradicts the facts proved in [16,17]. And in 1998, Pflug and Youssfi [10] proved that the minimal ball{z ∈ C n : |z| 2 + |zz | < 1} is not a Lu Qi-Keng domain if n 4, this negates the fact proved in [16].…”

“…In 1973, Matsuura [16] proved that any bounded complete circular domain with a center at the origin is a Lu Qi-Keng domain. In 1982, Kanemaru [17] proved that the domain D p = {z ∈ C n : |z 1 | 2/p1 + · · · + |z n | 2/pn < 1} is a Lu Qi-Keng domain.…”

“…In 1982, Kanemaru [17] proved that the domain D p = {z ∈ C n : |z 1 | 2/p1 + · · · + |z n | 2/pn < 1} is a Lu Qi-Keng domain. In 1979, Kakurai [18] used Theorem 1 in [16] proved that if D is itself a simply-connected bounded and minimal domain with a center at the origin, then D is the Lu Qi-Keng domain.…”

“…In fact, the above three results in [16][17][18] are not true. A domain is called complete circular if whenever it contains a point z, it also contains the one-dimensional disk {λz : |λ| 1}.…”