Some refined Schwarz-Pick lemmas (Q1396338)

The author studies some Schwarz-type theorems for holomorphic mappings \(f\:\mathbb B_n\rightarrow\mathbb B_m\), where \(\mathbb B_k\) denotes the unit Euclidean balls in \(\mathbb C^k\). The main result of the paper is the following theorem. If \(f(a)=b\) and \(f(A)=B\), then \[ \beta_m\biggl(Sf^\ast(a)\frac{\varphi_a(A)}{|\varphi_a(A)|},\;Tf^\ast(A)\frac{\varphi_A(a)}{|\varphi_A(a)|}\biggr)\leq 2\beta_n(a,A), \] where \(\beta_k\) denotes the Bergman distance for \(\mathbb B_k\), \(\varphi_c\:\mathbb B_n\rightarrow\mathbb B_n\) is the Möbius automorphism such that \(\varphi(0)=c\), \(\varphi(c)=0\), \(S\) and \(T\) are unitary isomorphisms of \(\mathbb C^m\) such that \(S(\varphi_b(B))=T(\varphi_B(b))\), and \(f^\ast(c):=\varphi_{f(c)}'(f(c))f'(c)\varphi_c'(0)\).