The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For an irreducible bounded symmetric domain $\Omega\subset \mathbb{C}^{d}$ in its Harish-Chandra realization, a positive integer number $m$ and a positive real number $\mu$, the Cartan-Hartogs domain $\Omega^{B^{m}}(\mu)$ is defined by

$ \begin{equation}\label{eq1.1} \Omega^{B^{m}}(\mu):=\left\{(z, w)\in \Omega\times \mathbb{C}^{m}\subset \mathbb{C}^{d}\times \mathbb{C}^{m}: \|w\|^2 < N_ \Omega (z, \overline{z})^{\mu} \right\}, \end{equation} $

(1.1)

where $\|\cdot\|$ is standard Hermitian norm in $\mathbb{C}^{m}$. Note $\Omega\times \{0\}\subset \Omega^{B^{m}}(\mu)$ and $b\Omega\times \{0\}\subset b\Omega^{B^{m}}(\mu)$ (where $bD$ denotes the boundary of the domain $D$). Obviously, each Cartan-Hartogs domain is a bounded complete circular domain.

Let $\Omega$ be an irreducible bounded symmetric domain in $\mathbb{C}^{d}$. Let Aut $(\Omega^{B^{m}}(\mu))$ be the holomorphic automorphism group of $\Omega^{B^{m}}(\mu)$. Let the family $G(\Omega^{B^{m}}(\mu))$ ( $\subset \mbox{ Aut}(\Omega^{B^{m}}(\mu))$) be exactly the set of all mappings $\Phi$ (see [16]):

$ \begin{equation}\label{1.2} \Phi(z, w)=\left(\varphi(z), U(w) \frac { N_\Omega(z_0, \bar z_0 )^{{\mu}/{2}}} {N_\Omega(z, \bar z_0)^{{\mu}}}\right) \end{equation} $

(1.2)

for $(z, w)\in \Omega^{B^{m}}(\mu)$, where $\varphi\in \mbox{ Aut}(\Omega)$, $U$ is a unitary transformation of $\mathbb{C}^{m}, $ and $z_0 =\varphi^{-1}(0).$ Then $G(\Omega^{B^{m}}(\mu))$ is a subgroup of $\mbox{ Aut}(\Omega^{B^{m}}(\mu))$ (see [1], Proposition 2.1). Obviously, as indicated in [16], every element of $G(\Omega^{B^{m}}(\mu))$ preserves the set $\Omega\times \{0\}\;(\subset \Omega^{B^{m}}(\mu))$ and $G(\Omega^{B^{m}}(\mu))$ is transitive on $\Omega\times \{0\}\;(\subset \Omega^{B^{m}}(\mu))$.

The Cartan-Hartogs domain $\Omega^{B^{m}}(\mu)$ is homogeneous if and only if $\Omega$ is the unit ball in $\mathbb{C}^{d}$ and $\mu=1$ (see [1], Lemma 3.1), that is, $\Omega^{B^{m}}(\mu)$ must be the unit ball in this case. Therefore, with the exception of the unit ball which is obviously homogeneous, each Cartan-Hartogs domain $\Omega^{B^{m}}(\mu)$ is a nonhomogeneous bounded domain. For the general reference of Cartan-Hartogs domains, see [1, 5-9, 11, 15-17], and references therein.

In 2012, by using the ball characterization theorem about noncompact automorphism groups (i.e., the Wong-Rosay theorem), Ahn-Byun-Park [1] proved the following theorem for irreducible bounded symmetric domains $\Omega$ of classical types.

Theorem 1.1(see [1])  Let $\Omega$ be an irreducible bounded symmetric domain. If the Cartan-Hartogs domain $\Omega^{B^{m}}(\mu)$ is not the unit ball, then $\mbox{ Aut}(\Omega^{B^{m}}(\mu))$ coincides with $G(\Omega^{B^{m}}(\mu))$(see (1.2) for the definition).

Remark  Let the Cartan-Hartogs domain $\Omega^{B^{m}}(\mu)$ be the unit ball. Since every element of $G(\Omega^{B^{m}}(\mu))$ preserves the set $\Omega\times \{0\}\;(\subset \Omega^{B^{m}}(\mu))$ and the unit ball is homogeneous, we have $G(\Omega^{B^{m}}(\mu))\subsetneqq \mbox{ Aut}(\Omega^{B^{m}}(\mu))$ in this case.

In 2006, Wang-Yin-Zhang-Roos [16] proved the following result.

Theorem 1.2(see [16])  Let $\Omega^{B^{m}}(\mu)$ be a Cartan-Hartogs domain. Let $X : \Omega^{B^{m}}(\mu)\rightarrow [0, 1)$ be the function defined by

$ \begin{equation}\label{e1} X=\frac{\|w\|^2}{N_{\Omega}(z, \overline{z})^{\mu}}. \end{equation} $

(1.3)

If $F$ is an automorphisms of $\Omega^{B^{m}}(\mu)$ with $X(F(z, w))$ $\equiv$ $X(z, w) $ on $\Omega^{B^{m}}(\mu)$, then $F\in G(\Omega^{B^{m}}(\mu))$.

In this paper, we prove the following conclusion.

Theorem 1.3  Let $\Omega_1, \Omega_2$ be two equidimensional irreducible bounded symmetric domains. Define

$ X_1(z, w):=\frac{\|w\|^2}{N_{\Omega_1}(z, \overline{z})^{\mu_1}}\;((z, w)\in \Omega_1^{B^{m}}(\mu_1)), \;\; X_2(z, w):=\frac{\|w\|^2}{N_{\Omega_2}(z, \overline{z})^{\mu_2}}\;((z, w)\in \Omega_2^{B^{m}}(\mu_2)) $

for Cartan-Hartogs domains $\Omega_1^{B^{m}}(\mu_1), \Omega_2^{B^{m}}(\mu_2)$, respectively. Let $\mathbf{U}\subset\Omega_1^{B^{m}}(\mu_1)$ be a neighborhood of the origin in $\Omega_1^{B^{m}}(\mu_1)$. Suppose that $F:\mathbf{U}\rightarrow \Omega_2^{B^{m}}(\mu_2)$ is a holomorphic mapping with $X_2(F(z, w))\equiv X_1(z, w)$ on $\mathbf{U}$. Then there exists a holomorphic automorphism $\Phi \in G(\Omega_2^{B^{m}}(\mu_2))$ such that $\Phi\circ F$ is the restriction on $\mathbf{U}$ of the standard linear isomorphism

$ \begin{equation}\label{1.3} \Phi_0(z, w)=(\mathcal{A}(z), U(w)), \end{equation} $

(1.4)

where $\mathcal{A}:\mathbb{C}^{d} \mapsto \mathbb{C}^{d}$ is a complex linear isomorphism of $\mathbb{C}^{d}$ with $\mathcal{A}(\Omega_1)=\Omega_2$, and $U$ is a unitary transformation of $\mathbb{C}^{m}$.

Theorem 1.3 obviously implies the following corollaries.

Corollary 1.4  Let $\Omega^{B^{m}}(\mu)$ be a Cartan-Hartogs domain. Let $X : \Omega^{B^{m}}(\mu)\rightarrow [0, 1)$ be the function defined by (1.3). If $F$ is a holomorphic self-mapping of $\Omega^{B^{m}}(\mu)$, then $F\in G(\Omega^{B^{m}}(\mu))$ if and only if $X(F(z, w))$ $\equiv$ $X(z, w) $ on $\Omega^{B^{m}}(\mu)$.

Combining Theorem 1.1 and Corollary 1.4, we immediately have the following result.

Corollary 1.5  Let $\Omega^{B^{m}}(\mu)$ be a Cartan-Hartogs domain. Let $X : \Omega^{B^{m}}(\mu)\rightarrow [0, 1)$ be the function defined by (1.3). Assume that $\Omega^{B^{m}}(\mu)$ is not the unit ball. If $F$ is a holomorphic self-mapping of $\Omega^{B^{m}}(\mu)$, then $F\in \mbox{ Aut}(\Omega^{B^{m}}(\mu))$ if and only if $X(F(z, w))$ $\equiv$ $X(z, w) $ on $\Omega^{B^{m}}(\mu)$.

The paper is organized as follows. In next section, we collect basic material about classical domains, and we will prove two lemmas which are necessary for the proof of Theorem 1.3. Finally Section 3 is dedicated to the proof of Theorem 1.3.

Let $\Omega$ be an irreducible bounded symmetric domain in $\mathbb{C}^d$ with the rank $r$ in its Harish-Chandra realization. The space of holomorphic polynomials on $\mathbb{C}^d$ can be decomposed into irreducible subspaces under the action of the isotropy group of the bounded symmetric domain $\Omega$ in $\mathbb{C}^d$. Let $\mathcal{K}$ be the connected component of the identity in the Lie group of the (complex linear) automorphisms of $\Omega$ leaving $0$ fixed. Under the action $f\mapsto f\circ k \; (k\in \mathcal{K})$ of $\mathcal{K}$, the space $\mathcal{P}$ of holomorphic polynomials on $\mathbb{C}^d$ admits the Peter-Weyl decomposition (see Th. 2.1 in [3] for references)

$ \begin{eqnarray*}\label{1.00} \mathcal{P}=\bigoplus_{\mathbf{m}}\mathcal{P}_{\mathbf{m}}, \end{eqnarray*} $

(2.1)

where the summation is taken over all partitions $\mathbf{m}:=(m_1, m_2, \cdots, m_r)$ of nonnegative integers such that $m_1\geq m_2 \geq \cdots \geq m_r \geq 0$, and the spaces $\mathcal{P}_{\mathbf{m}}$ are $\mathcal{K}$-invariant and irreducible. For each $\mathbf{m}$, we have $\mathcal{P}_{\mathbf{m}} \subset \mathcal{P}_{|\mathbf{m}|}$, where $\mathcal{P}_{|\mathbf{m}|}$ is the space of homogeneous holomorphic polynomials on $\mathbb{C}^d$ of degree $\mathbf{|m}|(:=\sum\limits_{j=1}^r m_j)$ (Obviously, $\mathcal{P}_{|\mathbf{m}|}$ is a $\mathcal{K}$-invariant subspace of $\mathcal{P}$). Let

$ \begin{equation}\label{1.4} {\langle}f, g {\rangle}:=\int_{\mathbb{C}^d}f(z)\overline{g(z)} e^{-m(z, \overline{z})} \frac{(\frac{\sqrt{-1}}{2\pi}\partial\overline{\partial}m(z, \overline{z}))^d}{d!} \end{equation} $

(2.2)

be an inner product on the space $\mathcal{P}$, where $m(z, \overline{z}):=-\left.\frac{\partial N_{\Omega}(tz, \overline{z})}{\partial t}\right|_{t=0}=zC_{\Omega} {\overline z}^t$ (where $C_{\Omega}$ is a positive definite Hermite matrix, see (2.10) for the details). For every partition $\mathbf{m}$, let $K_{\mathbf{m}}(z, \overline{z})$ be the reproducing kernel of $\mathcal{P}_{\mathbf{m}}$ with respect to (2.2). Since $\mathcal{P}_{\mathbf{m}} (\subset \mathcal{P}_{|\mathbf{m}|})$ is of finite demension, by definition $K_{\mathbf{m}}(\cdot, \cdot)$ is homogeneous holomorphic polynomials on $\mathbb{C}^d\times \mathbb{C}^d$ of bidegrees $(|\mathbf{m}|, |\mathbf{m}|)$.

Let $\Omega$ be an irreducible bounded symmetric domainin $\mathbb{C}^d$ with rank $r$ in its Harish-Chandra realization. Then there exists the Jordan triple product on $\mathbb{C}^d$ associated with the Bergman kernel of $\Omega$ and the space $\mathbb{C}^d$ endowed with the triple product is a simple Hermitian positive Jordan triple system (e.g., see Appendix A in [16]). Let $e_1, e_2, \cdots, e_r\in\mathbb{C}^d$ be a frame for $\mathbb{C}^d$. Then each $z\in \mathbb{C}^d$ has the spectral decomposition (see Th. Ⅵ. 2.3 and Def. Ⅵ. 2.2 in [4], p. 512-513, and Def. Ⅵ. 2.3 and Prop. Ⅵ. 2.6 in [4], p. 515-516)

$ \begin{equation}\label{1.5} z=k(z)\cdot(\lambda_1(z)e_1+\lambda_2(z)e_2+\cdots+\lambda_r(z)e_r), \end{equation} $

(2.3)

where $k(z)\in\mathcal{K}, \;\lambda_1(z)\geq \lambda_2(z)\geq \cdots \geq \lambda_r(z)\geq 0.$ The spectral norm of $z$ is defined by

$ \begin{equation}\label{1.6} \||z|\|:=\lambda_1(z). \end{equation} $

(2.4)

Then we have (see Def. Ⅵ.4.1 and Prop. Ⅵ.4.2 of [4], p. 524)

$ \begin{equation*} \Omega=\{z\in \mathbf{C}^d: \||z|\|<1\}. \end{equation*} $

Therefore, we have $1> \lambda_1(z)\geq \lambda_2(z)\geq \cdots \geq \lambda_r(z)\geq 0$ for any $z\in \Omega$.

The generic minimal polynomial of $\mathbb{C}^d$ (see Prop. Ⅵ. 2.6 and its proof in [4], p. 515-517)

$ \begin{equation*} m(t, z_1, \overline{z_2})=t^r-m_1(z_1, \overline{z_2})t^{r-1}+\cdots+(-1)^rm_r(z_1, \overline{z_2}) \end{equation*} $

satisfies

$ \begin{eqnarray*} m(t, z, \overline{z})=\prod\limits_{j=1}^r(t-\lambda^2_j(z)), \end{eqnarray*} $

where $m_1(\cdot, \cdot), $ $\cdots$, $m_r(\cdot, \cdot)$ on $\mathbb{C}^d\times \mathbb{C}^d$ are homogeneous holomorphic polynomials of, respectively, bidegrees $(1, 1)$, $\cdots$, $(r, r)$, $z=k(z)\cdot(\lambda_1(z)e_1+\lambda_2(z)e_2+\cdots+\lambda_r(z)e_r)$ is the spectral decomposition of $z$. The generic norm $N_\Omega$ is defined by

$ \begin{equation*} N_\Omega(z_1, \overline{z_2})= m(1, z_1, \overline{z_2}). \end{equation*} $

Then

$ \begin{eqnarray*}\label{1.8} N_\Omega(z, \overline{z})=\prod\limits_{j=1}^r(1-\lambda^2_j(z)). \end{eqnarray*} $

(2.5)

Note that, by definition, $N_\Omega(\cdot, \cdot)$ is a holomorphic polynomials on $\mathbb{C}^d\times \mathbb{C}^d$. The generic norm $N_\Omega$ is related to the kernels $K_{\mathbf{m}}$ by the formula (see Th. 3.8 in [3])

$ \begin{eqnarray*}\label{1.9} N_\Omega(z_1, \overline{z_2})^{-s}=\sum\limits_{\mathbf{m}}(s)_{\mathbf{m}}K_{\mathbf{m}}(z_1, \overline{z_2})\;\;(s\in \mathbb{C}, \; z_1, z_2\in \Omega), \end{eqnarray*} $

(2.6)

where the series converges uniformly and absolutely on compact subsets of $\Omega\times \Omega$, and $(s)_{\mathbf{m}}$ denotes the generalized Pochhammer symbol

$ \begin{eqnarray*}\label{1.10} (s)_{\mathbf{m}}:=\prod\limits_{j=1}^r\left(s-\frac{j-1}{2}a\right )_{m_j}, ~(x)_k=\frac{\Gamma(x+k)}{\Gamma(x)}=x(x+1)\cdots \cdots(x+k-1). \end{eqnarray*} $

(2.7)

By using logarithmic expansion, (2.5) implies the formula

$ \begin{eqnarray*}\label{e2.16} \ln N_\Omega(z, \overline{z})=-\sum\limits_{k=1}^{\infty}\frac{1}{k}\sum\limits_{j=1}^r\lambda_j^{2k}(z)\;\;(z\in \Omega). \end{eqnarray*} $

(2.8)

Using the spectral decomposition of $z$ in (2.3), $K_{\mathbf{m}}$ can be rewritten as (see Lemma 3.2 in [3], p. 235 for references)

$ \begin{eqnarray*}\label{1.7} K_{\mathbf{m}}(z, \overline{z})= K_{\mathbf{m}}(\sum\limits_{j=1}^r\lambda^2_j(z)e_j, \overline{e})\;\;(e=\sum\limits_{j=1}^re_j). \end{eqnarray*} $

Let

$ \begin{eqnarray*} e_k(t_1, \cdots, t_r)=\sum\limits_{1\leq i_1 < i_2 < \cdots < i_k\leq r}t_{i_1}t_{i_2}\cdots t_{i_r}~(1\leq k \leq r) \end{eqnarray*} $

be elementary symmetric polynomials. For a partition $\mathbf{m}=(1^{k_1}2^{k_2}\cdots r^{k_r})$, where $k_i$ is the number of parts of $\mathbf{m}$ that are equal to $i$ for $i \geq 1$. We define

$ \begin{eqnarray*} e_{\mathbf{m}}=\prod\limits_{j=1}^re_1^{k_1}e_2^{k_2}\cdots e_r^{k_2}. \end{eqnarray*} $

Then the set $\{e_{\mathbf{m}} :|\mathbf{m}|=n\}$ is a basis of a space consist of all symmetric polynomials of degree $n$ in variables $t_1, t_2, \cdots, t_r$, $n\in \mathbb{N}$ (see [13]} page 21). Thus, there exist constants $c_{\mathbf{m}}$ such that

$ \begin{eqnarray*} \sum\limits_{j=1}^rt_j^k=\sum\limits_{|\mathbf{m}|=k}c_{\mathbf{m}}e_{\mathbf{m}}(t_1, \cdots, t_r) \end{eqnarray*} $

for $k\geq 1, k\in \mathbb{N}$.

For $z=k\cdot(\lambda_1(z)e_1+\lambda_2(z)e_2+\cdots+\lambda_r(z)e_r)$, coefficients $m_k(z, \overline{z})$ of the generic minimal polynomial $m(t, z, \overline{z})$ may be written as

$ \begin{eqnarray*} m_k(z, \overline{z})=\sum\limits_{1\leq i_1 < i_2 < \cdots < i_k\leq r}\lambda^2_{i_1}(z)\lambda^2_{i_2}(z)\cdots \lambda^2_{i_r}(z)=e_k(\lambda^2_1(z), \lambda^2(z), \cdots, \lambda_r^2(z))~(1\leq k \leq r). \end{eqnarray*} $

Therefore

$ \begin{eqnarray*} \sum\limits_{j=1}^r\lambda_j^{2k}(z)=\sum\limits_{\mathbf{m}=(1^{k_1}2^{k_2}\cdots r^{k_r})\atop|\mathbf{m}|=\sum\limits_{j=1}^rjk_j=k}c_{\mathbf{m}}m_1^{k_1}(z, \overline{z})m_2^{k_2}(z, \overline{z})\cdots m_r^{k_r}(z, \overline{z}). \end{eqnarray*} $

This means that there exist constants $c_{\alpha\beta}$ such that

$ \begin{eqnarray*}\label{e2.17} \sum\limits_{j=1}^r\lambda_j^{2k}(z)=\sum\limits_{|\alpha|=|\beta|=k}c_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}, \end{eqnarray*} $

(2.9)

where $\alpha=(\alpha_1, \alpha_2, \cdots, \alpha_d)$, $\alpha_i\in \mathbb{N}, 1\leq i \leq d$, $|\alpha|=\sum\limits_{i=1}^d\alpha_i$ and $z^{\alpha}=\prod\limits_{i=1}^d z_i^{\alpha_i}$.

Let $z=k\cdot(\lambda_1(z)e_1+\lambda_2(z)e_2+\cdots+\lambda_r(z)e_r)$ be the spectral decomposition of $z$. For give $t\geq 0$, the spectral decomposition of $tz$ is $tz=k\cdot(t\lambda_1(z)e_1 +t\lambda_2(z)e_2+\cdots+t\lambda_r(z)e_r)$. From (2.5) and (2.6), we obtain

$\begin{eqnarray*} N_\Omega(tz, \overline{tz})=\prod\limits_{j=1}^r(1-t^2\lambda_j^2(z))=1-t^2\sum\limits_{j=1}^r\lambda_j^{2}(z)+\cdots\end{eqnarray*} $

and

$\begin{eqnarray*} N_\Omega(tz, \overline{tz})=\sum\limits_{\mathbf{m}}(-1)_{\mathbf{m}}K_{\mathbf{m}}(tz, \overline{tz}) =\sum\limits_{\mathbf{m}}(-1)_{\mathbf{m}}t^{2|\mathbf{m}|}K_{\mathbf{m}}(z, \overline{z})=1-t^2K_{(1, 0, \cdots, 0)}(z, \overline{z})+\cdots\end{eqnarray*} $

(Note the $r$-tuples $\mathbf{m}$ in (2.1) with $|\mathbf{m}|=1$ if and only if $\mathbf{m}=(1, 0, \cdots, 0)).$ Since $K_{(1, 0, \cdots, 0)}(\cdot, \cdot)$ is homogeneous holomorphic polynomials on $\mathbb{C}^d\times \mathbb{C}^d$ of bidegrees (1.1), then

$ \begin{eqnarray*}\label{e2.18} \sum\limits_{j=1}^r\lambda_j^{2}(z)=K_{(1, 0, \cdots, 0)}(z, \overline{z})\equiv zC_{\Omega}{\overline z}^t, \end{eqnarray*} $

(2.10)

where $C_{\Omega}$ is a Hermite matrix. Since $K_{(1, 0, \cdots, 0)}(z, \overline{z})\geq 0$ and $K_{(1, 0, \cdots, 0)}(z, \overline{z})=0$ iff $z=0$ by the definition, we get that $C_{\Omega}$ is a positive definite Hermite matrix.

Lemma 2.1  Let $\Omega_1$ and $\Omega_2$ be two irreducible bounded symmetric domains in $\mathbb{C}^d$ in their Harish-Chandra realization. Suppose that $\phi:\; \Omega_1\rightarrow \Omega_2$ is a holomorphic mapping such that

$ \begin{equation}\label{1.14} N_{\Omega_1}(z, \overline{z})\equiv N_{\Omega_2}(\phi(z), \overline{\phi(z)})^{\mu}\;\;(z\in \Omega_1), \end{equation} $

(2.11)

where $\mu>0$. Then $\phi$ is a biholomorphic mapping of $\Omega_1$ onto $\Omega_2$ with $\phi(0) = 0$ and $\mu=1$. Therefore, $\phi$ must be a complex linear automorphism of $\mathbb{C}^{d}$ with $\phi(\Omega_1)=\Omega_2$.

Proof  Let $z=(z_1, z_2, \cdots, z_d)$, $w=\phi(z)=(w_1(z), w_2(z), \cdots, w_d(z))$. Assume that $K_i$, $\gamma_i$ and $V_i$ are the Bergman kernel, the genus and the volume of $\Omega_i$ $(1\leq i \leq 2)$.

Combining $N(z, \overline{z})=0$ iff $z=0$ and (2.11), we obtain $\phi(0)=0$.

Using (2.11), we have

$ \begin{equation*} \left(\frac{\partial^2\ln N_{\Omega_1}}{\partial z_i\partial \overline{z_j}}(z, \overline{z})\right)_{1\leq i, j\leq d} =\mu {\phi}'(z) \left(\frac{\partial^2\ln N_{\Omega_2}} {\partial w_i\partial \overline{w_j}}(\phi(z), \overline{\phi(z)})\right)_{1\leq i, j\leq d} \overline{\phi '(z)}^{t}, \end{equation*} $

where

$ {\phi}'(z)=\left( \begin{array}{cccc} \frac{\partial w_1}{\partial z_1}&\frac{\partial w_2}{\partial z_1} &\cdots&\frac{\partial w_d}{\partial z_1} \\ \vdots&\vdots&\cdots&\vdots \\ \frac{\partial w_1}{\partial z_d}&\frac{\partial w_2}{\partial z_d} &\cdots&\frac{\partial w_d}{\partial z_d} \\ \end{array} \right). $

Since $K_i= {N_{\Omega_i}^{-\gamma_i}}/{V_i}$, we get

$ \frac{\partial^2\ln K_1}{\partial z_i\partial \overline{z_j}}=-\gamma_1\frac{\partial^2\ln N_{\Omega_1}}{\partial z_i\partial \overline{z_j}}, ~~\frac{\partial^2\ln K_2}{\partial w_i\partial \overline{w_j}}=-\gamma_2\frac{\partial^2\ln N_{\Omega_2}}{\partial w_i\partial \overline{w_j}}. $

Since $\left(\frac{\partial^2\ln K_1}{\partial z_i\partial \overline{z_j}}\right)_{1\leq i, j\leq d}$ and $\left(\frac{\partial^2\ln K_2}{\partial w_i\partial \overline{w_j}}\right)_{1\leq i, j\leq d}$ are positive definite Hermite matrices, we obtain that $\left(\frac{\partial^2\ln N_{\Omega_1}}{\partial z_i\partial \overline{z_j}} \right)_{1\leq i, j\leq d}$ and $\left(\frac{\partial^2\ln N_{\Omega_2}}{\partial w_i\partial \overline{w_j}}\right)_{1\leq i, j\leq d}$ are negative definite Hermite matrices. Therefore $\phi'(z)$ is an invertible matrix for any $z\in \Omega_1$.

Now we show that $\phi$ is a proper holomorphic mapping between $\Omega_1$ and $\Omega_2$. In fact, if there exists a sequence $\{p_j\}$ in $\Omega_1$ such that $p_j \rightarrow p_0 \in b\Omega_1$ ( $b\Omega$ stand for the boundary of $\Omega$ in $\mathbb{C}^d$) and $q_j=\phi (p_j)\rightarrow q_0 \in \Omega_2.$ Then (2.11) implies

$ N_{\Omega_1}(p_0, \overline{p_0})=N_{\Omega_2}(q_0, \overline{q_0})^{\mu}. $

This is a contradiction with $N_{\Omega_1}(p_0, \overline{p_0})=0, \; 0< N_{\Omega_2}(q_0, \overline{q_0})\leq 1.$ Therefore, $\phi$ must be a proper holomorphic mapping between $\Omega_1$ and $\Omega_2$.

Since the irreducible bounded symmetric domains $\Omega_1$ is simply connected, we have that $\phi$ is a biholomorphic mapping between $\Omega_1$ and $\Omega_2$. Since $\phi(0)=0$, from the Cartan's theorem, $\phi$ is a complex linear automorphism of $\mathbb{C}^d$ with $\phi(\Omega_1)=\Omega_2$.

Finally, we show that $\mu=1$. Since $\phi$ is a holomorphically isomorphism of $\Omega_1$ onto $\Omega_2$, we have $\Omega_1$ and $\Omega_2$ have the same genus $\gamma(:=\gamma_1=\gamma_2).$ Let $\phi(z)=zA \;(z\in \mathbb{C}^d)$ is the complex linear automorphism of $\mathbb{C}^d$ with $\phi(\Omega_1)=\Omega_2$ (where $A$ is an invertible $d\times d$ matrix). Then we have

$ V_2=|\det A|^2V_1, \;\;K_{1}(z, \bar z)=|\det A|^2K_{2}(zA, \overline{zA}). $

Thus, from $K_{i}(z, \bar z)= {N_{\Omega_i}(z, \bar z)^{ -\gamma}}/V_i\;\;( 1\leq i \leq 2), $ we get

$ N_{\Omega_1}(z, \bar z)\equiv N_{\Omega_2}(zA, \overline{zA})\; \;(z\in \Omega_1). $

Since $N_{\Omega_1}(z, \bar z)\;(z\in \Omega_1)$ takes any number in $(0, \;1]$, we have $\mu=1$ by (2.11). This proves Lemma 2.1.

Lemma 2.2  Let $\alpha=(\alpha_1, \cdots, \alpha_d)$ and $\beta=(\beta_1, \cdots, \beta_m)$ be tuples of non-negative integers. For $z\in \mathbb{C}^d, $ $w\in\mathbb{C}^m, $ set

$ \begin{eqnarray}\label{e2.1} &&P_{N}(z)=\sum\limits_{|\alpha|=N}d_{\alpha}z^{\alpha}, \end{eqnarray} $

(2.12)

$ \begin{eqnarray}\label{e2.2} &&Q_{N}(z, w)=\sum\limits_{|\alpha|+|\beta|=N\atop |\alpha|\geq 1, |\beta|\geq 1}e_{\alpha\beta}z^{\alpha}w^{\beta}, \end{eqnarray} $

(2.13)

$ \begin{eqnarray}\label{e2.3} &&R_{N}(z, w)=\sum\limits_{|\alpha|+|\beta|=N\atop |\alpha|\geq 1, |\beta|\geq 1}f_{\alpha\beta}z^{\alpha}w^{\beta}, \end{eqnarray} $

(2.14)

$ \begin{eqnarray}\label{e2.4} &&S_{2k}(z, \bar z)=\sum\limits_{|\alpha|=k\atop |\beta|=k}g_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}, \end{eqnarray} $

(2.15)

where $N\geq 1$, $k\geq 1$, $d_{\alpha}$ and $e_{\alpha\beta}$ are $d$-dimensional row vectors, $f_{\alpha\beta}$ are $m$-dimensional row vectors, and $g_{\alpha\beta}$ are complex numbers. Assume that $A$ is an invertible matrix of order $d$, $B$ is an invertible matrix of order $m$, and $\langle \cdot, \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb{C}^k$ ( $k=m$ or $d$).

(ⅰ) If

$ \begin{equation}\label{e2.9} 2{\rm Re}\langle wB, R_2(z, w)\rangle\equiv 0 \end{equation} $

(2.16)

and

$ \begin{equation}\label{e2.10} 2{\rm Re}\langle wB, R_3(z, w)\rangle+\|R_2(z, w)\|^2+ \|w\|^2S_2(z, \overline{z})\equiv 0, \end{equation} $

(2.17)

then

$ \begin{equation}\label{e2.11} R_2(z, w)\equiv 0, \; R_3(z, w)\equiv 0, \; S_2(z, \overline{z})\equiv 0. \end{equation} $

(2.18)

(ⅱ) Suppos that $N=2k+1$, $k\geq 2$ and

$ \begin{equation}\label{e2.5} 2{\rm Re}\langle wB, R_N(z, w)\rangle +2\mu\|w\|^2{\rm Re} \langle zA, P_{N-2}(z)+Q_{N-2}(z, w)\rangle +{\frac{1}{k}}\|w\|^2S_{2k}(z, \overline{z})\equiv 0, \end{equation} $

(2.19)

where $\|w\|^2\equiv \langle w, w\rangle :=\sum\limits_{j=1}^mw_j\overline{w_j}$. Then

$ \begin{equation}\label{e2.6} P_{N-2}(z)\equiv 0, \; Q_{N-2}(z, w)\equiv 0, \; R_{N}(z, w)\equiv 0, \; S_{2k}(z, \overline{z})\equiv 0. \end{equation} $

(2.20)

(ⅲ) Let $N=2k$ $(k\geq 2).$ If

$ \begin{equation}\label{e2.7} 2{\rm Re}\langle wB, R_N(z, w)\rangle +2\mu\|w\|^2{\rm Re}\langle zA, P_{N-2}(z)+Q_{N-2}(z, w)\rangle \equiv 0, \end{equation} $

(2.21)

then

$ \begin{equation}\label{e2.8} P_{N-2}(z)\equiv 0, \; Q_{N-2}(z, w)\equiv 0, \; R_{N}(z, w)\equiv 0. \end{equation} $

(2.22)

Proof  We only prove (2.20) here (the proof of (2.18) and (2.22) are the same as that of (2.20)).

Let $wB=\sum\limits_{j=1}^m\varepsilon_jw_j$, $zA=\sum\limits_{j=1}^d\eta_jz_j$, where $\{\varepsilon_j: 1\leq j\leq m\}$ and $\{\eta_j: 1\leq j \leq d\}$ are bases of $\mathbb{C}^m$ and $\mathbb{C}^d$, respectively. Since

$ \begin{eqnarray*}2{\rm Re}\langle wB, R_N(z, w)\rangle &=&\sum\limits_{1\leq j\leq m}\sum\limits_{|\alpha|+|\beta|=N\atop |\alpha|\geq 1, |\beta|\geq 1} \left\{\langle \varepsilon_j, f_{\alpha\beta}\rangle w_j\overline{z}^{\alpha}\overline{w}^{\beta}+ \langle f_{\alpha\beta}, \varepsilon_j\rangle \overline{w_j} z^{\alpha}w^{\beta}\right\}, \\ &=& \|w\|^2{\rm Re}\langle zA, P_{N-2}(z)+Q_{N-2}(z, w)\rangle \\ &=& \|w\|^2\sum\limits_{1\leq j\leq d} \left\{ \sum\limits_{|\alpha|=N-2} \left( \langle \eta_j, d_{\alpha}\rangle z_j\overline{z}^{\alpha}+\langle d_{\alpha}, \eta_j\rangle z^{\alpha}\overline{z_j} \right)\right. \\ && \left. + \sum\limits_{|\alpha|+|\beta|=N-2\atop |\alpha|\geq 1, |\beta|\geq 1}\left( \langle \eta_j, e_{\alpha\beta} \rangle z_j\overline{z}^{\alpha}\overline{w}^{\beta}+\langle e_{\alpha\beta}, \eta_j\rangle z^{\alpha}w^{\beta}\overline{z_j}\right) \right\}, \\ \|w\|^2S_{2k}(z, \overline{z})&=&\|w\|^2\sum\limits_{|\alpha|=k\atop |\beta|=k}g_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}, \end{eqnarray*} $

and sets $\{z^{\alpha}, \overline{z}^{\alpha}:1\leq |\alpha|\leq N-1\}$, $\{z_j\overline{z}^{\alpha}, \overline{z_j} z^{\alpha}: 1\leq j \leq d, |\alpha|=N-2\}$, $\{z_j\overline{z}^{\alpha}, \overline{z_j}z^{\alpha}: 1\leq j \leq d, 1\leq |\alpha|\leq N-3\}$ and $\{z^{\alpha}\overline{z}^{\beta}: |\alpha|=|\beta|=k(>1)\}$ are pairwise disjoint, (2.19) implies

$ \begin{eqnarray}\label{e2.12} &&\sum\limits_{1\leq j\leq m}\sum\limits_{1\leq |\beta|=N-|\alpha|}<\varepsilon_j, f_{\alpha\beta}>w_j\overline{w}^{\beta}\equiv 0\quad (1\leq |\alpha|\leq N-1), \end{eqnarray} $

(2.23)

$ \begin{eqnarray}\label{e2.13} &&<\eta_j, d_{\alpha}>=0\quad (1\leq j \leq d, |\alpha|=N-2), \end{eqnarray} $

(2.24)

$ \begin{eqnarray}\label{e2.14} &&\sum\limits_{1\leq |\beta|=N-2-|\alpha|}<\eta_j, e_{\alpha\beta}>\overline{w}^{\beta}\equiv 0\quad (1\leq j \leq d, 1\leq|\alpha|\leq N-3), \end{eqnarray} $

(2.25)

$ \begin{eqnarray}\label{e2.15} &&g_{\alpha\beta}=0\quad (|\alpha|=k, |\beta|=k). \end{eqnarray} $

(2.26)

Therefore, (2.23) implies

$ <\varepsilon_j, f_{\alpha\beta}>=0\quad (1\leq j \leq m, |\alpha|+|\beta|=N, |\alpha|\geq 1, |\beta|\geq 1). $

Since $\{\varepsilon_j: 1\leq j\leq m\}$ is a basis of $\mathbb{C}^m$, we have

$ f_{\alpha\beta}=0\quad ( |\alpha|+|\beta|=N, |\alpha|\geq 1, |\beta|\geq 1), $

by (2.14), we get

$ R_{N}(z, w)\equiv 0. $

Similarly, from (2.24), (2.25) and (2.26), we have $P_{N-2}(z)\equiv 0, \; Q_{N-2}(z, w)\equiv 0, \; S_{2k}(z, \overline{z})\equiv 0.$ This proves Lemma 2.2.

Proof  We divide our proof into four steps.

(ⅰ) Let $F(z, w)=(F_1(z, w), F_2(z, w))$. Then, from $X_2\circ F\equiv X_1$ on $\mathbf{U}$, we have

$ \frac{\|F_2(z, w)\|^2}{N_ {\Omega_2}(F_1(z, w), \overline{F_1(z, w)})^{\mu_2}} \equiv \frac{\|w\|^2}{N_ {\Omega_1} (z, \bar z)^{\mu_1}}\;\; ((z, w)\in \mathbf{U}). $

Thus we have $F_2(z, 0)=0, (z, 0)\in \mathbf{U}$, and so $F(0, 0)=(\widetilde{u}_0, 0)(\in\Omega_2\times \{0\})$. Therefore, there exists $\Phi\in G(\Omega_2^{B^{m}}(\mu_2))$ with $\Phi\circ F(0, 0)=(0, 0)$ (Note $\Phi$ leaves the function $X_2$ on $\Omega_2^{B^{m}}(\mu_2)$ invariant). Let $H:\;=\Phi\circ F.$ Then

$ H:\;\mathbf{U}\rightarrow \Omega_2^{B^{m}}(\mu_2) $

is a holomorphic mapping with $X_2(H(z, w))\equiv X_1(z, w)$ on $\mathbf{U}$ and $H(0, 0)=(0, 0)$.

Write $H(0, w)$ in the following form

$\begin{eqnarray*} H(0, w)\equiv (h_1(w), h_2(w))=(wV+\sum\limits_{j\geq 2}f_j(w), wU+\sum\limits_{j\geq 2}g_{j}(w))(\in \Omega_2^{B^{m}}(\mu_2)), (0, w)\in \mathbf{U}, \end{eqnarray*} $

where all components of $f_j(w)$ and $g_j(w)$ are homogeneous polynomials of degree $j \;(j\geq 2)$. For $(0, w)\in \mathbf{U}$ (i.e., $w\in B^m$), there exists a positive number $\delta_w$ such that $(0, tw)\in \mathbf{U}$, $\forall t\in [0, \delta_w]$. By $X_2\circ H(0, tw)\equiv X_1(0, tw)$, we have

$ \begin{eqnarray*} \left\|wU+\sum\limits_{j\geq 2}t^{j-1}g_{j}(w)\right\|^2\equiv \|w\|^2\left({N_{\Omega_2}\left(twV+\sum\limits_{j\geq 2}t^jf_j(w), \overline{twV+\sum\limits_{j \geq 2}t^jf_j(w)}\right)}\right)^{\mu_2}. \end{eqnarray*} $

Take $t\rightarrow 0^{+}, $ we get

$ \begin{equation*} \|wU\|^2\equiv \|w\|^2, \end{equation*} $

that is, $U$ is a unitary matrix of order $m$.

From

$ \begin{equation}\label{e3.1} \|h_2(w)\|^2\equiv \|w\|^2\left(N_{\Omega_2}\left(h_1(w), \overline{h_1(w)}\right)\right)^{\mu_2}, (0, w)\in \mathbf{U}, \end{equation} $

(3.1)

we get

$ \|h_2(w)\|\leq \|w\|, (0, w)\in \mathbf{U}. $

For $\zeta\in \mathbb{C}^m, \|\zeta\|=1$, there exists a positive number $\eta_\zeta$ such that $(0, \lambda\zeta)\in \mathbf{U}$ for all $|\lambda|\leq \eta_\zeta$. We define

$ g(\lambda):= < h_2(\lambda \zeta), \zeta U>, |\lambda|\leq \eta_\zeta. $

Then $g(0)=0, g'(0)=1$ and $ |g(\lambda)|\leq |\lambda| $ for $|\lambda|\leq \eta_\zeta$.

Let

$ \widetilde{g}(\lambda):=\left\{\begin{array}{cc} \frac{g(\lambda)}{\lambda}, &0<|\lambda|\leq \eta_\zeta, \\ 1, &\lambda=0. \end{array} \right. $

Then $\widetilde{g}$ is a holomorphic map on $\{\lambda\in \mathbb{C}:|\lambda|<\eta_\zeta\}$, and by $\|h_2(w)\|\leq \|w\|, (0, w)\in \mathbf{U}$, we have $|\widetilde{g}(\lambda)|\leq 1$. Since $\widetilde{g}(0)=1$, according to maximum modulus principle, it follows that $\widetilde{g}\equiv 1$, thus $g(\lambda)=\lambda, |\lambda|\leq \eta_\zeta$, that is

$ <\lambda^{-1}h_2(\lambda \zeta), \zeta U> =1, 0<|\lambda|\leq \eta_\zeta. $

Using $\|h_2(\lambda \zeta)\|\leq \lambda $ and $\|\zeta U\|=1$, we get $h_2(\lambda \zeta)=\lambda\zeta U, |\lambda|\leq \eta_\zeta$. Thus $h_2(w)=wU, (0, w)\in \mathbf{U}$.

Owing to (3.1), we get

$ \begin{equation*} N_{\Omega_2}\left(h_1(w), \overline{h_1(w)}\right)\equiv 1, (0, w)\in \mathbf{U}, \end{equation*} $

that is, $h_1(w)\equiv 0, (0, w)\in \mathbf{U}$.

Let $H(z, w)=(H_1(z, w), H_2(z, w))$. Then we have

$ H(0, 0)=(0, 0), \; H_2(z, 0)\equiv 0, \;H_1(0, w)\equiv 0, \;H_2(0, w)=wU, (z, 0)\in \mathbf{U}, (0, w)\in \mathbf{U}, $

where $U$ is a unitary matrix of order $m$. This means

$ \begin{eqnarray*} H_1(z, w)=zA+\sum\limits_{j\geq 2}(P_j(z)+Q_j(z, w)), \quad H_2(z, w)=wU+\sum\limits_{j\geq 2}R_{j}(z, w), (z, w)\in \mathbf{U}, \end{eqnarray*} $

where $P_j, Q_j$ and $R_j$ are homogeneous polynomials of degree $j$, which are given by (2.12), (2.13) and (2.14) respectively.

(ⅱ) For $(z, w)\in \mathbf{U}$ with $w\neq 0$, there exists a positive number $\delta_{z, w} $ such that $(tw, tw)\in \mathbf{U}$ for all $t\in [0, \delta_{z, w}]$. Since $X_2\circ H(tz, tw)=X_1(tz, tw)$ $(\forall t\in [0, \delta_{z, w}])$, it follows

$ \begin{equation}\label{e2.19} \begin{array}{l} \left\|\frac{1}{\|w\|}wU+\frac{1}{\|w\|}\sum\limits_{j\geq 2}t^{j-1}R_{j}(z, w)\right\|^2 \\\\ =\displaystyle \frac{{N_{\Omega_2}\left(tzA+\sum\limits_{j\geq 2}t^j(P_j(z)+Q_j(z, w)), \overline{tzA+\sum\limits_{j\geq 2}t^j(P_j(z)+Q_j(z, w))}\right)}^{\mu_2}}{N_{\Omega_1}(tz, t\overline{z})^{\mu_1}}. \end{array} \end{equation} $

(3.2)

By (2.8) we obtain

$ \begin{eqnarray}\label{e2.20} &&\ln{N_{\Omega_2}\left(tzA+\sum\limits_{j\geq 2}t^j(P_j(z)+Q_j(z, w)), \overline{tzA+\sum\limits_{j\geq 2}t^j(P_j(z)+Q_j(z, w))}\right)} \nonumber\\ &=&-t^2\left(\sum\limits_{j=1}^{r_2}\Lambda^2_j(zA+\sum\limits_{j\geq 2}t^{j-1}(P_j(z)+Q_j(z, w)))+o(1)\right), \end{eqnarray} $

(3.3)

$ \begin{eqnarray}\label{e2.21} &&\ln{N_{\Omega_1}(tz, t\overline{z})}=-t^2\left(\sum\limits_{j=1}^{r_1}\lambda^2_j(z)+o(1)\right), \end{eqnarray} $

(3.4)

where $r_1$ and $r_2$ are the ranks of $\Omega_1$ and $\Omega_2$, $z\in \Omega_1$ has the spectral decomposition $z=k(z)\cdot(\lambda_1(z)e_1+\lambda_2(z)e_2+\cdots+\lambda_r(z)e_r)$ and $u\in \Omega_2$ has the spectral decomposition $u=\tilde k (u)\cdot(\Lambda_1(u)\tilde e_1+\Lambda_2(u) \tilde e_2+\cdots+\Lambda_r(u)\tilde e_r)$ (Note $\lambda_j(tz)=t \lambda_j(z)$ for $t\geq 0$ here). By substituting (3.3) and (3.4) into (3.2), for $t\in [0, \delta_{z, w}]$, we have

$ \begin{eqnarray}\label{e2.22} &&\ln{\left(1+2{\rm Re} < wU, R_2(z, w)>\frac{t}{\|w\|^2}+2{\rm Re} < wU, R_3(z, w)>\frac{t^2}{\|w\|^2}\right.}\nonumber\\ &&\left.+\|R_2(z, w)\|^2\frac{t^2}{\|w\|^2}+o(t^2)\right)\nonumber\\ &=&-t^2\left(\mu_2\sum\limits_{j=1}^{r_2}\Lambda_j^2(zA+\sum\limits_{j\geq 2}t^{j-1}(P_j(z)+Q_j(z, w)))-\mu_1\sum\limits_{j=1}^{r_1}\lambda_j^2(z)+o(1)\right). \end{eqnarray} $

(3.5)

Dividing the two sides of the equation (3.5) by $t^2$ and taking $t\rightarrow 0^{+}$, we get

$ \begin{eqnarray*} &&2{\rm Re} < wU, R_2(z, w)>\equiv 0, \\ &&2{\rm Re} < wU, R_3(z, w)>+\|R_2(z, w)\|^2+ \|w\|^2S_2(z, \overline{z})\equiv 0, \end{eqnarray*} $

where

$ \begin{eqnarray*}\label{e2.23} S_{2k}(z, \overline{z}):=\mu_2\sum\limits_{j=1}^{r_2}\Lambda_j^{2k}(zA)-\mu_1\sum\limits_{j=1}^{r_1}\lambda_j^{2k}(z), \end{eqnarray*} $

(3.6)

in view of (2.9), there exist constants $g_{\alpha\beta}$ such that

$ \begin{eqnarray*} S_{2k}(z, \overline{z})=\sum\limits_{|\alpha|=|\beta|=k}g_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}. \end{eqnarray*} $

By Lemma 2.2 and (2.10), we have

$ \begin{equation}\label{e2.23.1} R_2(z, w)\equiv 0, ~R_3(z, w)\equiv 0, ~S_2(z, \overline{z})\equiv 0, ~\mu_2AC_{\Omega_2}\overline{A}^{t}=\mu_1C_{\Omega_1}. \end{equation} $

(3.7)

Since $C_{\Omega_1}$ and $C_{\Omega_2}$ are positive definite Hermite matrices and $\mu_2AC_{\Omega_2}\overline{A}^{t}=\mu_1C_{\Omega_1}$, we get that $A$ is an invertible matrix of order $d$.

(ⅲ) Now we show that for all $j>3$, $P_{j-2}(z)\equiv 0, Q_{j-2}(z, w)\equiv 0, R_j(z, w)\equiv 0, S_{2\left[\frac{j-1}{2}\right]}\equiv 0$ by the reduction to absurdity. Let

$ \begin{equation}\label{e.23.2} N:=\min\left\{j: P_{j-2}(z) \not\equiv 0, \; Q_{j-2}(z, w) \not\equiv 0, \;R_j(z, w) \not\equiv 0 ~\text{or}~ S_{2\left[\frac{j-1}{2}\right]} \not\equiv 0\right\}. \end{equation} $

(3.8)

From (3.7), we know $N\geq 4$. Now assume $N<+\infty$ here.

Using (3.2), we have

$ \begin{eqnarray}\label{e2.24} &&\left\|\frac{1}{\|w\|}wU+\frac{1}{\|w\|}\sum\limits_{j\geq N}t^{j-1}R_{j}(z, w)\right\|^2 \nonumber\\ &=&\frac{{N_2\left(tzA+\sum\limits_{j\geq N-2}t^j(P_j(z)+Q_j(z, w)), \overline{tzA+\sum\limits_{j\geq N-2}t^j(P_j(z)+Q_j(z, w))}\right)}^{\mu_2}}{N_1(tz, t\overline{z})^{\mu_1}}. \end{eqnarray} $

(3.9)

By (2.8) and (2.10), we get

$ \begin{eqnarray}\label{e2.25} \ln{N_2\left(tzA+\sum\limits_{j\geq N-2}t^j(P_j(z)+Q_j(z, w)), \\ \overline{tzA+\sum\limits_{j\geq N-2}t^j(P_j(z)+Q_j(z, w))}\right)} \nonumber\\ =-t^2\sum\limits_{j=1}^{r_2}\Lambda_j^2(zA+\sum\limits_{l\geq N-2}t^{l-1}(P_l(z)+Q_l(z, w))\\ -\sum\limits_{k=2}^{\left[\frac{N-1}{2}\right]}\frac{t^{2k}}{k}\sum\limits_{j=1}^{r_2}\Lambda_j^{2k}(zA)+o(t^{N-1}) \nonumber\\ =-2t^{N-1}Re < zAC_{\Omega_2}, P_{N-2}(z)+Q_{N-2}(z, w)>\\ -\sum\limits_{k=1}^{\left[\frac{N-1}{2}\right]} \frac{t^{2k}}{k}\sum\limits_{j=1}^{r_2}\Lambda_j^{2k}(zA)+o(t^{N-1}) \end{eqnarray} $

(3.10)

and

$ \begin{eqnarray*}\label{e2.26} \ln{N_1(tz, t\overline{z})}=-\sum\limits_{k=1}^{\left[\frac{N-1}{2}\right]}\frac{t^{2k}}{k}\sum\limits_{j=1}^{r_1}\lambda_j^{2k}(z)+o(t^{N-1}). \end{eqnarray*} $

(3.11)

Substituting (3.10) and (3.11) into (3.9), we obtain for all $t\in [0, \delta_{z, w}]$,

$ \begin{eqnarray}\label{e2.27} \ln{\left(1+2{\rm Re} < wU, R_N(z, w)>\frac{t^{N-1}}{\|w\|^2}+o(t^{N-1})\right)} \nonumber\\ =-\left\{2\mu_2t^{N-1}{\rm Re} < zAC_{\Omega_2}, P_{N-2}(z)+Q_{N-2}(z, w)>\\ +\sum\limits_{j=1}^{\left[\frac{N-1}{2}\right]}\frac{t^{2j}}{j}S_{2j}(z, \overline{z})+o(t^{N-1})\right\}\nonumber\\ =-\left\{2\mu_2t^{N-1}{\rm Re} < zAC_{\Omega_2}, P_{N-2}(z)+Q_{N-2}(z, w)>\right.\nonumber\\ \left.+\frac{t^{2\left[\frac{N-1}{2}\right]}}{\left[\frac{N-1}{2}\right]}S_{2\left[\frac{N-1}{2}\right]}(z, \overline{z})+o(t^{N-1})\right\}, \end{eqnarray} $

(3.12)

where $S_{2k}(z, \overline{z})$ is the same as (3.6).

When $N=2k+1$, by (3.12), we have

$ \begin{equation*} \begin{array}{l} 2{\rm Re} < wU, R_N(z, w)>+2\mu_2\|w\|^2{\rm Re} < zAC_{\Omega_2}, P_{N-2}(z)\\ +Q_{N-2}(z, w)>+{\frac{1}{k}}\|w\|^2S_{2k}(z, \overline{z})\equiv 0. \end{array} \end{equation*} $

By Lemma 2.2, we obtain

$ \begin{equation*} P_{N-2}(z)\equiv 0, ~Q_{N-2}(z, w)\equiv 0, ~R_{N}(z, w)\equiv 0, ~S_{2\left[\frac{N-1}{2}\right]}(z, \overline{z})\equiv S_{2k}(z, \overline{z})\equiv 0. \end{equation*} $

This is the contradiction with (3.8).

When $N=2k$, by (3.12), we get

$ \begin{eqnarray*} &&2{\rm Re} < wU, R_N(z, w)>+2\mu_2\|w\|^2{\rm Re} < zAC_{\Omega_2}, P_{N-2}(z)+Q_{N-2}(z, w)>\equiv 0, \\ &&S_{2\left[\frac{N-1}{2}\right]}(z, \overline{z})\equiv 0. \end{eqnarray*} $

From Lemma 2.2 we have

$ \begin{equation*} P_{N-2}(z)\equiv 0, Q_{N-2}(z, w)\equiv 0, R_{N}(z, w)\equiv 0. \end{equation*} $

This is also the contradiction with (3.8).

(ⅳ) From (ⅰ), (ⅱ) and (ⅲ), we get

$ \Phi\circ F(z, w)\equiv H(z, w)=(zA, wU), N_1(z, \overline{z})^{\mu_1}=N_2(zA, \overline{zA})^{\mu_2}, (z, w)\in \mathbf{U} $

and

$ \begin{equation*} F(z, w)=\left(\psi(zA), \frac{N_2(u_0, \overline{u_0})^{\frac{\mu_2}{2}}}{N_2(zA, \overline{u_0})^{\mu_2}}wU\right), (z, w)\in \mathbf{U}. \end{equation*} $

Let $D:=\{z\in \Omega_1:(z, w)\in \mathbf{U}\}$. Then $D$ is a neighborhood of the origin in $\Omega_1$. By $T(z, w):=N_1(z, w)^{\frac{\mu_1}{\mu_2}}-N_2(zA, w\overline{A})$ is a holomorphic function on $\Omega_1\times \Omega_1$ and $T(z, w)\equiv 0$ on $D\times D$, we obtain $T(z, w)\equiv 0$ on $\Omega_1\times \Omega_1$. Thus

$ N_1(z, \overline{z})^{\mu_1}=N_2(zA, \overline{zA})^{\mu_2}, z\in \Omega_1. $

By Lemma 2.1, we have that mapping $\mathcal{A}: z\in \Omega_1\mapsto zA\in \Omega_2$ is a biholomorphic mapping of $\Omega_1$ onto $\Omega_2$ with $\mu_1=\mu_2$. So $\Phi_0(z, w):=(zA, wU)$ is a the standard linear isomorphism of $\Omega_1^{B^m}(\mu_1)$ into $\Omega_1^{B^m}(\mu_2)$. Therefore $\Phi\circ F$ is the restriction on $\mathbf{U}$ of a biholomorphic mapping $\Phi_0$.

Let $\phi(z)=\psi(zA)$, $z_0=u_0A^{-1}$, then $\phi$ is a biholomorphic mapping of $\Omega_1$ onto $\Omega_2$ with $\phi(z_0)=0$, thus

$ \begin{equation*} F(z, w)=\left(\phi(z), \frac{N_1(z_0, \overline{z_0})^{\frac{\mu_1}{2}}}{N_1(z, \overline{z_0})^{\mu_1}}wU\right), (z, w)\in \mathbf{U}. \end{equation*} $

This proves Theorem 1.3.