Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds and asymptotic expansions for the radii of convexity and uniform convexity of normalized Bessel functions

Árpád Baricz, Pranav Kumar, Sanjeev Singh

TL;DR

This work addresses the large-order behavior of the radii of convexity and uniform convexity for normalized Bessel functions. It develops a framework based on Weierstrass factorization of Dini functions, Rayleigh sums, and the Laguerre–Pólya class, and leverages Euler–Rayleigh inequalities and ordinary potential polynomials to obtain both sharp bounds and explicit asymptotic expansions for the radii of convexity and uniform convexity of the normalized forms $\,g_\nu$ and $\,h_\nu$. The authors establish leading constants $c\approx0.535898$ and $d\approx1.17157$ governing the convexity radii (with higher-order corrections given by recurrence-derived coefficients) and analogous constants $\tilde{c}\approx0.298438$ and $\tilde{d}\approx0.627719$ for the uniform convexity radii, together with recurrences for the correction terms and Euler–Rayleigh bounds. They also provide generalized bounds in terms of potential polynomials and present numerical illustrations validating the asymptotic expansions and showing that, for fixed large $\nu$, the uniform convexity radius is smaller than the convexity radius. These results yield computable, accurate approximations for the radii and deepen the understanding of the geometric properties of normalized Bessel functions in the large-order regime, with implications for zeros, wave propagation, and related asymptotic analyses.

Abstract

This paper explores the asymptotic behaviour of the radii of convexity and uniform convexity for normalized Bessel functions with respect to large order. We provide detailed asymptotic expansions for these radii and establish recurrence relations for the associated coefficients. Additionally, we derive generalized bounds for the radii of convexity and uniform convexity by applying the Euler-Rayleigh inequality and potential polynomials. The asymptotic inversion method and Rayleigh sums are the main tools used in the proofs.

Bounds and asymptotic expansions for the radii of convexity and uniform convexity of normalized Bessel functions

TL;DR

This work addresses the large-order behavior of the radii of convexity and uniform convexity for normalized Bessel functions. It develops a framework based on Weierstrass factorization of Dini functions, Rayleigh sums, and the Laguerre–Pólya class, and leverages Euler–Rayleigh inequalities and ordinary potential polynomials to obtain both sharp bounds and explicit asymptotic expansions for the radii of convexity and uniform convexity of the normalized forms and . The authors establish leading constants and governing the convexity radii (with higher-order corrections given by recurrence-derived coefficients) and analogous constants and for the uniform convexity radii, together with recurrences for the correction terms and Euler–Rayleigh bounds. They also provide generalized bounds in terms of potential polynomials and present numerical illustrations validating the asymptotic expansions and showing that, for fixed large , the uniform convexity radius is smaller than the convexity radius. These results yield computable, accurate approximations for the radii and deepen the understanding of the geometric properties of normalized Bessel functions in the large-order regime, with implications for zeros, wave propagation, and related asymptotic analyses.

Abstract

This paper explores the asymptotic behaviour of the radii of convexity and uniform convexity for normalized Bessel functions with respect to large order. We provide detailed asymptotic expansions for these radii and establish recurrence relations for the associated coefficients. Additionally, we derive generalized bounds for the radii of convexity and uniform convexity by applying the Euler-Rayleigh inequality and potential polynomials. The asymptotic inversion method and Rayleigh sums are the main tools used in the proofs.
Paper Structure (5 sections, 12 theorems, 205 equations, 4 figures)

This paper contains 5 sections, 12 theorems, 205 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary and main results
  3. Asymptotic expansions for the radii of convexity of normalized Bessel functions
  4. Bounds and asymptotic expansions for the radii of uniform convexity of normalized Bessel functions
  5. Proofs of the preliminary and main results

Key Result

Lemma 1

For any positive integer $k$ and positive real $\nu>k$, the Rayleigh sum in eta_def has the convergent Laurent expansion where for any fixed non negative integer $n$, the coefficients $\eta_n^{(k)}$ can be evaluated by the recurrence relation and $a_n^{(k)}$ is given by

Figures (4)

  • Figure 1: The image of the open disk $\mathbb{D}_{r}$ under the Bessel function $z\mapsto g_\nu(z),$ where $r\sim 5.208\ldots$ is the approximative value of the radius of convexity of $g_\nu(z)$ considering the first two terms of \ref{['conv_num_val']} for $\nu=50$.
  • Figure 2: The image of the open disk $\mathbb{D}_{r}$ under the Bessel function $z\mapsto h_\nu(z),$ where $r\sim 59.437\ldots$ is the approximative value of the radius of convexity of $h_\nu(z)$ considering the first two terms of \ref{['h_conv_num_val']} for $\nu=50$.
  • Figure 3: The image of the open disk $\mathbb{D}_{r}$ under the Bessel function $z\mapsto g_\nu(z),$ where $r\sim 3.891\ldots$ is the approximative value of the radius of uniform convexity of $g_\nu(z)$ considering the first two terms of \ref{['unif_num_val']} for $\nu=50$.
  • Figure 4: The image of the open disk $\mathbb{D}_{r}$ under the Bessel function $z\mapsto h_\nu(z),$ where $r\sim 31.86\ldots$ is the approximative value of the radius of uniform convexity of $h_\nu(z)$ considering the first two terms of \ref{['h_unif_conv_num_val']} for $\nu=50$.

Theorems & Definitions (32)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • ...and 22 more