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Weighted contractivity for derivatives of functions in the Bergman space on the unit disk

David Kalaj, Petar Melentijević

TL;DR

This work generalizes sharp derivative-type inequalities for Bergman-space functions on the unit disk by introducing weighted measures $d\mu_{n,\alpha}$ and $d\nu_{n,\alpha}$ and establishing $n$-th derivative bounds via hypergeometric functions, hyperbolic isoperimetric geometry, and a convexity argument on superlevel sets. It proves two main sharp bounds: $R_{n,\mu}(f,\Omega)\le 1-(1+s/\pi)^{1-X}$ with $X=(n+1)(n+2+\alpha)$ for $1\le n\le 3$, and $R_{n,\nu}(f,\Omega)\le 1-(1+s/\pi)^{1-2n-(\alpha+2)}$ for all $n\ge 1$, without attainment of equality. A key lemma (Lemma 2.1) is established via Parseval-type arguments for $d\nu_{n,\alpha}$ and a Jacobi-polynomial framework for $d\mu_{n,\alpha}$ (valid for $n\le 3$), underpinning the derivative estimates. Finally, a limiting procedure recovers Kalaj's Fock-space inequality from the Bergman-space results, with Laguerre polynomials appearing in the limit, thus connecting hyperbolic-Gaussian models to Euclidean Fock-space theory.

Abstract

In a recent paper, Ramos and Tilli proved certain sharp inequality for analytic functions in subdomains of the unit disk. We will generalize their main inequality for derivatives of functions from Bergman space with respect to two diferent measures. Some connections with an analog for the Fock spaces, earlier investigated in Kalaj, will also be discussed.

Weighted contractivity for derivatives of functions in the Bergman space on the unit disk

TL;DR

This work generalizes sharp derivative-type inequalities for Bergman-space functions on the unit disk by introducing weighted measures and and establishing -th derivative bounds via hypergeometric functions, hyperbolic isoperimetric geometry, and a convexity argument on superlevel sets. It proves two main sharp bounds: with for , and for all , without attainment of equality. A key lemma (Lemma 2.1) is established via Parseval-type arguments for and a Jacobi-polynomial framework for (valid for ), underpinning the derivative estimates. Finally, a limiting procedure recovers Kalaj's Fock-space inequality from the Bergman-space results, with Laguerre polynomials appearing in the limit, thus connecting hyperbolic-Gaussian models to Euclidean Fock-space theory.

Abstract

In a recent paper, Ramos and Tilli proved certain sharp inequality for analytic functions in subdomains of the unit disk. We will generalize their main inequality for derivatives of functions from Bergman space with respect to two diferent measures. Some connections with an analog for the Fock spaces, earlier investigated in Kalaj, will also be discussed.
Paper Structure (5 sections, 7 theorems, 118 equations)

This paper contains 5 sections, 7 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Bergman spaces on $\mathbb{C}^+$ and $\mathbb{D}$
  3. Proof of Theorem \ref{['kalaj']}
  4. A proof of Lemma 2.1
  5. Consequences to Fock space case

Key Result

Theorem 1.1

Let $\alpha>-1,$ and $s>0$ be fixed. Among all functions $f\in {\mathcal{A}_\alpha}$ and among all measurable sets $\Omega\subset \mathbb{D}$ such that $\mu(\Omega)=s$, the quotient $R(f,\Omega)$ as defined in eq:optimal-bergman-object satisfies the inequality where $\mathbb{D}_s$ is a disc centered at the origin with $\mu(\mathbb{D}_s) = s.$ Moreover, there is equality in eq:upper-bound-quotient

Theorems & Definitions (10)

  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.1
  • proof