[Paper Review] On a theorem of Landau and Toeplitz

Robert B. Burckel, Donald E. Marshall+1 more|ArXiv.org|Mar 24, 2006

Analytic and geometric function theory4 references3 citations

TL;DR

This paper provides a streamlined proof of the Landau-Toeplitz theorem, which bounds the derivative at zero of analytic functions on the unit disk by the diameter of their image, and extends it to include quantitative estimates on the growth of the maximum modulus. The key result establishes that equality in the derivative bound occurs if and only if the function is linear with unimodular derivative.

ABSTRACT

The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of Carathéodory, who attributed it to Erhard Schmidt. Since then, Schwarz's Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a streamlined proof of this result and also extend it to include bounds on the growth of the maximum modulus.

Motivation & Objective

• To provide a concise and accessible proof of the Landau-Toeplitz theorem, which generalizes Schwarz's Lemma by replacing the maximum modulus with the diameter of the image set.
• To extend the theorem by including quantitative bounds on the growth of the maximum modulus of analytic functions under diameter constraints.
• To investigate the equality case in the derivative bound, showing that equality implies linearity of the function.
• To explore quantitative versions of the equality condition, aiming for explicit estimates on the deviation from linearity.
• To pose and initiate investigation into related geometric extremal problems involving hyperbolic centers and radii of convex domains with bounded diameter.

Proposed method

• Decompose the analytic function $ f $ into its odd and even parts: $ f(z) = f_o(z) + f_e(z) $, where $ f_o(z) = (f(z) - f(-z))/2 $.
• Apply Schwarz’s Lemma to the odd part $ f_o $, which satisfies $ |f_o(z)| \leq 1 $ and $ f_o(0) = 0 $, to deduce $ |f^{lat}(0)| \leq 1 $.
• Use the diameter condition $ \operatorname{Diam}f(\mathbb{D}) \leq 2 $ to show that the diameter of $ f(r\mathbb{D}) $ grows linearly as $ D_r = 2r $ for $ 0 < r < 1 $.
• Define auxiliary functions $ h_u(z) = (f(z) - f(-uz))/z $ for $ u \in \mathbb{T} $, and show that $ D_r / r $ is constant and equal to 2, implying $ D_r = 2r $.
• Use the function $ g_w(z) = (f(z) - f(-w))/(2f^\prime(0)) $, which fixes $ w $ and preserves the disk $ |z| \leq r $, to deduce $ \operatorname{Im} g_w^\prime(w) = 0 $, leading to $ f^\prime(z)/(2f^\prime(0)) $ being constant.
• Conclude that $ f(z) = f(0) + f^\prime(0)z $, proving the equality case.

Experimental results

Research questions

• RQ1 Under what conditions does equality hold in the Landau-Toeplitz derivative bound, and what does this imply about the structure of the analytic function?
• RQ2 Can the Landau-Toeplitz theorem be extended to provide quantitative estimates on the deviation of $ f(z) - f(0) - f^\prime(0)z $ from zero in terms of $ |f^\prime(0)| $?
• RQ3 What is the maximal hyperbolic radius $ R_h(\Omega) $ of a convex domain $ \Omega $ with diameter 2 and hyperbolic center density $ \Lambda(\Omega) \leq m $ for $ m > 1 $?
• RQ4 Given a function $ f $ with $ \operatorname{Diam}f(\mathbb{D}) \leq 2 $, what is the supremum of $ M(f) = \min_w \sup_z |f(z) - f(w)| $ under the constraint $ |f^\prime(w_f)|(1 - |w_f|^2) \geq a $ for $ a < 1 $?

Key findings

• The Landau-Toeplitz theorem is proven with a streamlined argument using the odd part of $ f $, showing $ |f^\prime(0)| \leq 1 $ whenever $ \operatorname{Diam}f(\mathbb{D}) \leq 2 $.
• Equality in $ |f^\prime(0)| = 1 $ holds if and only if $ f(z) = a + cz $ for some $ a, c \in \mathbb{C} $ with $ |c| = 1 $, proving the rigidity of the extremal case.
• The diameter of the image $ f(r\mathbb{D}) $ grows linearly: $ \operatorname{Diam}f(r\mathbb{D}) = 2r $ for all $ 0 < r < 1 $, under the equality condition.
• An explicit quantitative bound is derived: $ \max_{|z| < r} |f(z) - f(0) - f^\prime(0)z| \leq (1 - |f^\prime(0)|) \phi(r) $, where $ \phi(r) $ is an explicit function, addressing Problem 6.1.
• For functions with $ \operatorname{Diam}f(\mathbb{D}) \leq 2 $, the image diameter of $ f(r\mathbb{D}) $ is realized on the boundary $ r\mathbb{T} $, due to the open mapping theorem.
• If equality holds in the diameter-based bound for the $ n $-th coefficient, then $ f(z) = g(z^n) $ with $ g $ linear, implying $ f(z) = c_n z^n $ up to a constant, as shown in Corollary 3.2.