# 微积分网课代修|函数代写Function theory代考|MATH824 The Schwarz Lemma at the Boundary

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## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|The Schwarz Lemma at the Boundary

There has been interest for some time in studying Schwarz lemmas at the boundary of a domain. Löwner conducted early studies of a result weaker than the one presented here; his motivation was the study of distortion theorems. Our methods are quite distinct from Löwner’s (see [VEL] for a consideration of classical results with references). The standard reference for this new material is the paper $[\mathrm{BUK}]$ of Burns and Krantz.
Theorem 2.6.1. Let $\varphi: D \rightarrow D$ be a holomorphic function such that
$$\varphi(\zeta)=1+(\zeta-1)+\mathcal{O}\left(|\zeta-1|^{4}\right)$$
as $\zeta \rightarrow 1$. Then $\varphi(\zeta) \equiv \zeta$ on the disk
Compare this result with the uniqueness part of the classical Schwarz lemma. In that context, we assume that $\varphi(0)=0$ and $\left|\varphi^{\prime}(0)\right|=1$. At the boundary we must work harder. In fact, the following example shows that the size of the error term cannot be reduced from fourth order to third order.
Example 2.6.2. The function
$$\varphi(\zeta)=\zeta-\frac{1}{10}(\zeta-1)^{3}$$
satisfies the hypotheses of the theorem with the exponent 4 replaced by 3 . Yet clearly this $\varphi$ is not the identity.

To verify this example, we need only check that $\varphi$ maps the disk to the disk. It is useful to let $\zeta=1-\tau$ and consider therefore the function
$$\widetilde{\varphi}(\tau)=1-\tau+\left[\frac{1}{10} \tau^{3}\right] .$$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Normal Families

The concept of “normal family” is one of the most elegant and most powerful in complex function theory. It hinges on several key ideas: (i) the topology on the space of holomorphic functions, (ii) the Cauchy estimates, (iii) the Ascoli-Arzelà theorem. When properly viewed, Montel’s theorem is really just the triangle inequality formulated in certain invariant metrics.

The applications of normal families are manifold. Of course they are used decisively in the modern proof of the Riemann mapping theorem. They are used in the proof of Picard’s theorems, especially the great Picard theorem. They are used to study automorphisms of domains. And they can be used in the estimation of invariant metrics.

In the present chapter we shall explore all of these avenues, and imbue the reader with a greater appreciation for normal families.

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考| The Schwarz Lemma at the Boundary

$$\varphi(\zeta)=1+(\zeta-1)+\mathcal{O}\left(|\zeta-1|^{4}\right)$$

$$\varphi(\zeta)=\zeta-\frac{1}{10}(\zeta-1)^{3}$$

$$\widetilde{\varphi}(\tau)=1-\tau+\left[\frac{1}{10} \tau^{3}\right] .$$

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考| Normal Families

“正则族”的概念是复杂函数理论中最优雅、最有力的概念之一。它取决于几个关键思