MATHEMATICS

P. RUSEV

ANALYTIC MAPPINGS AND THE BERGMAN FUNCTION

(Presented by Academician M. A. Lavrent'ev on 2 III 1964)

Let \(G\) be an arbitrary bounded domain in the space \(C^n\) of complex variables \(z_1,\ldots,z_n\). By \(L(G,z^0)\) \((z^0\in G)\) denote the set of all analytic functions \(f(z)=f(z_1,\ldots,z_n)\), regular and single-valued in \(G\) and, moreover, satisfying the condition \(|f(z^0)|=1\). The Bergman function \(K_G(z^0;\bar z^0)\) of the domain \(G\) at the point \(z^0\), as is known, is defined by the equality

\[ \{K_G(z^0;\bar z^0)\}^{-1}=\inf_{f\in L(G,z^0)} A_G[f(z)], \]

where

\[ A_G[f(z)]=\int_G |f(z)|^2\,d\omega . \]

Every collection \((\varphi_1,\ldots,\varphi_n)\) of \(n\) analytic functions \(\varphi_k(z)\) \((k=1,2,\ldots,n)\), regular in the domain \(G\) (possibly multiple-valued), such that the function

\[ J(z,\varphi_1,\ldots,\varphi_n)= \frac{\partial(\varphi_1,\ldots,\varphi_n)} {\partial(z_1,\ldots,z_n)} \]

is regular and single-valued in \(G\), will be called an analytic mapping of the domain \(G\). The analytic mapping \((\varphi_1,\ldots,\varphi_n)\) is called normalized at the point \(z^0\in G\) if \(|J(z^0,\varphi_1,\ldots,\varphi_n)|=1\). By \(M(G,z^0)\) denote the set of all analytic mappings of the domain \(G\) normalized at the point \(z^0\in G\). Put

\[ \{P_G(z^0)\}^{-1}= \inf_{(\varphi_1,\ldots,\varphi_n)\in M(G,z^0)} A_G[J(z,\varphi_1,\ldots,\varphi_n)]. \]

Theorem 1. For every point \(z^0\in G\) the equality

\[ K_G(z^0;\bar z^0)=P_G(z^0) \]

holds.

Denote by \(E(G)\) the set of those points \(z^0\in G\) for which the following is fulfilled: whatever analytic mapping \((\varphi_1,\ldots,\varphi_n)\in M(G,z^0)\) may be, one always has \(A_G[J(z,\varphi_1,\ldots,\varphi_n)]\geq V(G)\), where \(V(G)=A_G\) is the volume of the domain \(G\) \((^1)\).

Theorem 2. In order that a point \(z^0\in E(G)\), it is necessary and sufficient that the equality

\[ K_G(z^0;\bar z^0)=\{V(G)\}^{-1} \]

hold.

Proof. Suppose that \(K_G(z^0;\bar z^0)=\{V(G)\}^{-1}\). If an analytic mapping \((\varphi_1,\ldots,\varphi_n)\in M(G,z^0)\), then

\[ A_G[J(z,\varphi_1,\ldots,\varphi_n)]\geq \{P_G(z^0)\}^{-1}= \{K_G(z^0;\bar z^0)\}^{-1}=V(G). \]

Let \(z_0\in E(G)\). Suppose that \(K_G(z^0;\bar z^0)>\{V(G)\}^{-1*}\). Then for every analytic mapping \((\varphi_1,\ldots,\varphi_n)\in M(G,z^0)\)

\[ A_G[J(z,\varphi_1,\ldots,\varphi_n)]\geq V(G), \]

and, consequently,

\[ \{P_G(z^0)\}^{-1}\geq V(G)>\{K_G(z^0;\bar z^0)\}^{-1}. \]

The last inequality contradicts Theorem 1.

Theorem 3. Let \(G=G_1\times\cdots\times G_n\); \(G_m\) \((m=1,2,3,\ldots,n)\) are bounded simply connected domains of the space \(C^1\). If the set \(E(G)\) contains at least one point \(z^0=(z_1^0,\ldots,z_n^0)\), then \(G\) is a polycylinder with center at the point \(z^0\), and the set \(E(G)\) consists only of the point \(z^0\).

\[ \text{* Note that always } K_G(z^0;\bar z^0)\geq \{V(G)\}^{-1}. \]

Proof. If one takes into account the equalities
\[ V(G)=\prod_{m=1}^{n} V(G_m) \]
and
\[ K_G(z^0;\bar z^0)=\prod_{m=1}^{n} K_{G_m}(z_m^0;\bar z_m^0), \]
then it is necessary to prove Theorem 3 for \(n=1\). Let the point \(z_1^0\in E(G_1)\), and let \(\zeta=\varphi(z)\) be a function satisfying the conditions \(\varphi(z_1^0)=0\), \(\varphi'(z_1^0)=1\), and mapping \(G_1\) one-to-one onto the disk \(\Delta_{z_1^0}: |\zeta|<R_{z_1^0}\), where \(R_{z_1^0}\) is the conformal radius of the domain \(G_1\) at the point \(z_1^0\). As is known,
\[ K_{G_1}(z_1^0;\bar z_1^0)=\{\pi R_{z_1^0}^{2}\}^{-1} \]
\(([^1],\) p. 90), and consequently
\[ V(G_1)=\pi R_{z_1^0}^{2}. \]
It follows that \(\varphi^{-1}(\zeta)=z_1^0+\zeta\), i.e., the domain \(G\) is a disk with center at the point \(z_1^0\) and radius \(R_{z_1^0}\). If one takes into account that for the disk \(\Delta_1: |z_1|<R\) the Bergman function is
\[ K_{\Delta_1}(z_1^0;\bar z_1^0)=\pi^{-1}R^{2}\bigl(R^{2}-|z_1^0|^{2}\bigr)^{-2}, \]
one may conclude that the set \(E(\Delta_1)\) consists only of the point \(z_1^0=0\).

Remark. It follows from Theorem 3 that if the domain \(G\subset C^n\) is a polycylinder with center at the point \(z^0\), then for every analytic mapping \((\varphi_1,\ldots,\varphi_n)\in M(G,z^0)\) the inequality
\[ A_G[J(z,\varphi_1,\ldots,\varphi_n)]\ge V(G) \]
holds.* Moreover, as is easy to show, the equality sign is possible if and only if
\[ |J(z,\varphi_1,\ldots,\varphi_n)|\equiv 1. \]
We note that if \(n>1\), it does not follow from the last equality that the analytic mapping \((\varphi_1,\ldots,\varphi_n)\) is “linear.”

Theorem 4. Let the functions \(\varphi_1,\ldots,\varphi_n\) be regular and single-valued in the domain \(G\subset C^n\), and let \(T=(\varphi_1,\ldots,\varphi_n)\) be a one-to-one analytic mapping belonging to the set \(M(G,z^0)\). Put \(H=T(G)\), and let the point \(z^0\in E(G)\). Then, if \(V(H)=V(G)\), the point \(Tz^0=w^0\in E(H)\).

Let \(T=(\varphi_1,\ldots,\varphi_n)\) be a one-to-one analytic mapping of the polycylinder \(G\subset C^n\), belonging to the set \(M(G,z^0)\), where \(z^0\) is the center of \(G\). Put \(H=T(G)\), and suppose that \(V(H)=V(G)\). By Theorem 4, the point \(w^0=Tz^0\in E(H)\). Moreover, if \(H\) is a polycylindrical domain, then it will necessarily be a polycylinder (Theorem 3), and then, according to a theorem of H. Cartan (2, p. 30), the mapping \(T\) is linear.

Thus, we have established the following theorem:

Theorem 5. Let \(G\subset C^n\) be a polycylinder with center at the point \(z^0\), and let \(T=(\varphi_1,\ldots,\varphi_n)\in M(G,z^0)\) be a one-to-one analytic mapping of the domain \(G\) onto a polycylindrical domain \(H\). If \(V(H)=V(G)\), then \(H\) is a polycylinder with center at the point \(w^0=Tz^0\), and the mapping \(T\) is linear.

Remark. The last assertion, together with the remark to Theorem 3, is in a certain sense an analogue of the so-called inner theorem of areas from the theory of functions of one complex variable. We note that Theorem 5 can be proved by relying only on the above-mentioned theorem of H. Cartan and on the remark to Theorem 3.

The author expresses his gratitude to Prof. B. V. Shabat for useful comments on the present work.

Mathematical Institute
with Computing Center
of the Bulgarian Academy of Sciences

Received
26 II 1964

REFERENCES

* This assertion can also be proved directly.

1. B. A. Fuks, Special Chapters in the Theory of Analytic Functions of Several Complex Variables, Moscow, 1963. ↩

2. H. Cartan, J. Math. pures et appl., 10 (1931). ↩