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Remarks on Redheffer's inequality

Nagi Suzuki, Shingo Takeuchi

TL;DR

The paper investigates how Redheffer's inequality and its Zhu–Sun generalization can be used to study the first eigenvalue $\lambda(p)$ of the $p$-Laplacian on $[-1,1]$, establishing an alternative proof of its monotonicity and refining its lower bound. It further extends Redheffer-type inequalities to generalized trigonometric functions, deriving a cosine-type bound for $\cos_{p,q}$ and broadening the inequality framework to a wider class of antiperiodic functions. The results connect classical inequality theory with nonlinear spectral problems, yielding sharper eigenvalue estimates and expanding the applicability of Redheffer-type bounds to generalized trigonometric structures. These contributions enhance understanding of $p$-Laplacian spectra and provide tools for analyzing generalized sinusoidal functions in nonlinear settings.

Abstract

Redheffer's inequality and its extensions are applied to study the behavior and estimates of the first eigenvalue of $p$-Laplacian with respect to $p$. Furthermore, a Redheffer-type inequality for the generalized trigonometric function is extended to a broader class.

Remarks on Redheffer's inequality

TL;DR

The paper investigates how Redheffer's inequality and its Zhu–Sun generalization can be used to study the first eigenvalue of the -Laplacian on , establishing an alternative proof of its monotonicity and refining its lower bound. It further extends Redheffer-type inequalities to generalized trigonometric functions, deriving a cosine-type bound for and broadening the inequality framework to a wider class of antiperiodic functions. The results connect classical inequality theory with nonlinear spectral problems, yielding sharper eigenvalue estimates and expanding the applicability of Redheffer-type bounds to generalized trigonometric structures. These contributions enhance understanding of -Laplacian spectra and provide tools for analyzing generalized sinusoidal functions in nonlinear settings.

Abstract

Redheffer's inequality and its extensions are applied to study the behavior and estimates of the first eigenvalue of -Laplacian with respect to . Furthermore, a Redheffer-type inequality for the generalized trigonometric function is extended to a broader class.
Paper Structure (6 sections, 11 theorems, 47 equations)

This paper contains 6 sections, 11 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Alternative proof of the monotonicity of $\lambda(p)$
  5. Improvement to the estimate of $\lambda(p)$
  6. Extension of Redheffer's inequality to generalized trigonometric functions

Key Result

Proposition 3.1

$\lambda(p)$ is strictly increasing with respect to $p$ in $(1,\infty)$.

Theorems & Definitions (14)

  • Proposition 3.1: KTT2017
  • Lemma 3.2
  • Proposition 3.3: H1997BD2012
  • Proposition 3.4: KT2025
  • Theorem 3.5
  • Lemma 3.6: ZS2008
  • proof : Proof of Theorem \ref{['thm:eigenvalue']}
  • Remark 3.7
  • Proposition 3.8: OT2021
  • Proposition 3.9
  • ...and 4 more