(a) Prove a "Schwarz's lemma" for harmonic functions. If u...

(a) Prove a "Schwarz's lemma" for harmonic functions. If $u$ is real valued and harmonic in $\mathbb{D}$ with $|u| \leq 1$ and $u(0)=0$, then $$ |u(z)| \leq \frac{4}{\pi} \arctan |z| $$ When can equality hold? (b) Formulate and prove an upper bound for $u(z)$, where $u$ is harmonic on $\mathbb{D}$ with $u(0)=$ 0 and $u \leq 1$. Hint: Use a conformal map.

(a) Prove a "Schwarz's lemma" for harmonic functions. If $u$ is real valued and harmonic in $\mathbb{D}$ with $|u| \leq 1$ and $u(0)=0$, then
$$
|u(z)| \leq \frac{4}{\pi} \arctan |z|
$$
When can equality hold?
(b) Formulate and prove an upper bound for $u(z)$, where $u$ is harmonic on $\mathbb{D}$ with $u(0)=$ 0 and $u \leq 1$.
Hint: Use a conformal map.

Key Concepts

Conformal Mapping

Conformal mappings are functions that preserve angles and local shapes, allowing one to transform complex domains into more manageable ones. In the context of harmonic functions, these mappings are useful for transporting problems from the unit disk to domains where the boundary behavior is simpler or better understood, thereby enabling the derivation of uniform bounds or optimal estimates.

Maximum Principle

The maximum principle is a fundamental concept in the study of harmonic functions, asserting that the maximum and minimum values of a non-constant harmonic function on a closed bounded domain occur on the boundary. This principle is crucial in developing bounds and uniqueness results, as it ensures that interior values are controlled by the boundary data, a key insight when proving inequalities like the harmonic Schwarz lemma.

Harmonic Functions

Harmonic functions are solutions to Laplace's equation, meaning they satisfy an important second-order partial differential equation that models equilibrium states in various physical contexts. Their defining properties, including the mean value property and the maximum principle, facilitate the derivation of bounds and uniqueness results, making them central to potential theory and complex analysis.

Schwarz's Lemma (and its Harmonic Analogue)

Originally formulated for holomorphic functions, Schwarz's lemma provides a sharp bound on the magnitude of functions vanishing at the origin. Its analogue for harmonic functions extends this idea, showing that, under suitable normalization conditions, harmonic functions are bounded in magnitude by a function (in this example, involving the arctan) of the distance from the origin. This result connects the geometry of the domain with the behavior of the function, highlighting the interplay between analytic structure and harmonicity.

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