| Summary: | 碩士 === 東海大學 === 數學系 === 89 === Let $\Gamma$ denote the set of symmetrized bidisc. In this thesis
we discuss the Schwarz lemma on $\Gamma$ also known as the special
flat problem on $\Gamma$ as:
Given
$\alpha_{2}\in\mathbb{D},~\alpha_{2}\neq0~$
and $(s_{2},p_{2})\in\Gamma$, find an analytic function
$\varphi:\mathbb{D}\rightarrow\Gamma$with
$\varphi(\lambda)=(s(\lambda),p(\lambda))$ satisfies
$$\varphi(0)=(0,0),~\varphi(\alpha_{2})=(s_{2},p_{2})$$
Based on the equality of Carath\'odory and Kobayashi distances,
and the Schur's theorem, we construct an analytic function
$\varphi$ to solve this problem.
Keywords: Spectral Nevanlinna-Pick interpolation,
Poincar\'{e} distance, Carath\'odory distance, Kobayashi
distance, Symmetrized bidisc, Schwarz lemma.
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