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Optimal universal growth for integral means of normalized logarithmic derivatives in the Carathéodory class

Yixin He, Quanyu Tang, Teng Zhang

Abstract

We determine the optimal universal growth scale for the integral means of normalized logarithmic derivatives in the Carathéodory class. This resolves a problem of D.~K.~Thomas.

Optimal universal growth for integral means of normalized logarithmic derivatives in the Carathéodory class

Abstract

We determine the optimal universal growth scale for the integral means of normalized logarithmic derivatives in the Carathéodory class. This resolves a problem of D.~K.~Thomas.
Paper Structure (5 sections, 6 theorems, 74 equations)

This paper contains 5 sections, 6 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. An H2-estimate for the nonconstant part of Log p
  3. A uniform upper bound and a fixed-function refinement
  4. Sharpness on the power scale
  5. Optimality of the universal gauge

Key Result

Theorem 1.2

For every $p\in\mathcal{P}$ and $0<r<1$, Moreover, for each fixed $p\in\mathcal{P}$,

Theorems & Definitions (13)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:upper']}
  • proof : Proof of Theorem \ref{['thm:sharp']}
  • ...and 3 more