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Note on the second derivative of bounded analytic functions

Gangqiang Chen

Abstract

Assume $z_0$ lies in the open unit disk $\mathbb{D}$ and $g$ is an analytic self-map of $\mathbb{D}$. We will determine the region of values of $g''(z_0)$ in terms of $z_0$, $g(z_0)$ and the hyperbolic derivative of $g$ at $z_0$, and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for $|g''(z_0)|$ than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Sz{á}sz's inequality, which provides a sharp upper bound for $|g''(z_0)|$ depending only on $|z_0|$.

Note on the second derivative of bounded analytic functions

Abstract

Assume lies in the open unit disk and is an analytic self-map of . We will determine the region of values of in terms of , and the hyperbolic derivative of at , and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Sz{á}sz's inequality, which provides a sharp upper bound for depending only on .
Paper Structure (3 sections, 7 theorems, 65 equations)

This paper contains 3 sections, 7 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Variability region of the second derivative
  3. Estimates of the second derivative

Key Result

Theorem A (The Schwarz-Pick Lemma)

Let $z_0, \delta_0\in \mathbb{D}$. Suppose that $g\in \mathcal{H}$, $g(z_0)=\delta_0$. Set Then the region of values of $g'(z_0)$ is the closed disk and $g(z)$ is the form of $T_{\delta_0}(T_{-z_0}(z) g^*(z))$, where $g^*\in\mathcal{S}$. Further, $g'(z_0)\in \partial\mathbb{D}(0, \dfrac{1-|\delta_0|^2}{1-|z_0|^2})$ if and only if $g(z)=g_{\alpha}(z)$ for some constant $\alpha \in \partial \mathb

Theorems & Definitions (11)

  • Theorem A (The Schwarz-Pick Lemma)
  • Theorem B (Dieudonné's Lemma)
  • Lemma 2.1: Yamashita $Yamashita1994$
  • Theorem 2.2
  • proof
  • Corollary 2.3: The Second-Order Dieudonné's Lemma
  • Theorem 3.1
  • proof : Proof.
  • Remark 3.2
  • Corollary 3.3: szasz1920
  • ...and 1 more