[Paper Review] Geometric and monotonic properties of hyper-Bessel functions

İbrahim Aktaş, Árpád Baricz+1 more|arXiv (Cornell University)|Feb 14, 2018

Analytic and geometric function theoryMathematics31 references23 citations

TL;DR

This paper investigates geometric and monotonic properties of normalized hyper-Bessel functions, focusing on the radii of starlikeness, convexity, and uniform convexity. Using Euler-Rayleigh inequalities and the Laguerre-Pólya class of entire functions, the authors derive explicit transcendental equations and tight bounds for these radii, extending classical results from Bessel functions to the hyper-Bessel setting with precise inequalities for the first positive zero and related functions.

ABSTRACT

Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler-Rayleigh inequalities and Laguerre-P\'{o}lya class of real entire functions, intensively.

Motivation & Objective

• To extend geometric function theory results—previously known for classical Bessel functions—to the broader class of hyper-Bessel functions.
• To determine the radii of starlikeness, convexity, and uniform convexity for normalized hyper-Bessel functions.
• To establish sharp bounds for the first positive zero of the normalized hyper-Bessel function and related quantities.
• To analyze monotonicity and interlacing properties of the zeros using infinite product and sum representations.
• To derive Redheffer-type inequalities and bounds for the hyper-Bessel function using the Laguerre-Pólya class and Euler-Rayleigh sums.

Proposed method

• Utilizes the infinite sum and product representations of the normalized hyper-Bessel function $ f_{\alpha_d}(z) = z J_{\alpha_d}(z) $, derived from generalized hypergeometric functions.
• Applies Euler-Rayleigh inequalities to the zeros of $ f_{\alpha_d}(z) $, using the first and second elementary symmetric sums of the reciprocals of the $ (d+1) $-th powers of the zeros.
• Employs the Laguerre-Pólya class to analyze the real entire function structure and prove interlacing and monotonicity properties of the zeros.
• Derives differential inequalities for the logarithmic derivatives of $ f_{\alpha_d} $ to characterize the radii of starlikeness, convexity, and uniform convexity.
• Uses the monotonicity of power series ratios and the monotone form of l’Hospital’s rule to prove that the ratio $ \Sigma_{\alpha_d}(x) = \frac{\log J_{\alpha_d}(x)}{\log \left(1 - \frac{x^{d+1}}{j_{\alpha_d,1}^{d+1}} \right)} $ is decreasing.
• Establishes Redheffer-type inequalities and bounds for $ J_{\alpha_d}(x) $ by analyzing the absolutely monotonic behavior of the logarithmic derivative of $ J_{\alpha_d}(x) $.

Experimental results

Research questions

• RQ1 What are the exact transcendental equations satisfied by the radii of starlikeness, convexity, and uniform convexity of normalized hyper-Bessel functions?
• RQ2 How do the bounds for the first positive zero of the normalized hyper-Bessel function compare to known bounds for classical Bessel functions?
• RQ3 What is the relationship between the zeros of the normalized hyper-Bessel function and the zeros of its derivative, and how does this relate to interlacing properties?
• RQ4 Can Redheffer-type inequalities and bounds for the hyper-Bessel function be derived using the Laguerre-Pólya class and Euler-Rayleigh sums?
• RQ5 To what extent can the geometric and monotonic properties of classical Bessel functions be naturally extended to hyper-Bessel functions?

Key findings

• The radius of starlikeness $ r^\star(f_{\alpha_d}) $ is the smallest positive root of $ z J'_{\alpha_d}(z) + J_{\alpha_d}(z) = 0 $, with bounds $ \sqrt[d+1]{\frac{(d+1)^d A(\alpha_d)}{d+1}} < r^\star(f_{\alpha_d}) < \sqrt[d+1]{\frac{(d+1)^d A(\alpha_d)}{d+1}} $, where $ A(\alpha_d) = \prod_{i=1}^d (\alpha_i + 1) $.
• The radius of convexity $ r_c(f_{\alpha_d}) $ is the smallest positive root of $ (d+1) z J'_{\alpha_d}(z) + z^2 J''_{\alpha_d}(z) + J_{\alpha_d}(z) = 0 $, and satisfies $ \sqrt[d+1]{\frac{(d+1)^{d+1} A(\alpha_d)}{d+2}} < (r_c(f_{\alpha_d}))^{d+1} < \frac{(d+2)(d+1)^{d+1} A(\alpha_d) B(\alpha_d)}{(d+2)^2 B(\alpha_d) - (2d+3) A(\alpha_d)} $, where $ B(\alpha_d) = \prod_{i=1}^d (\alpha_i + 2) $.
• The radius of uniform convexity $ r_{uc}(f_{\alpha_d}) $ is the smallest positive root of $ 2z^2 J''_{\alpha_d}(z) + 5z J'_{\alpha_d}(z) + J_{\alpha_d}(z) = 0 $, and is characterized by the unique solution in $ (0, \psi_{\alpha_d,1}) $ to $ 1 + 2z \frac{f''_{\alpha_d}(z)}{f'_{\alpha_d}(z)} = 0 $.
• The first positive zero $ j_{\alpha_d,1} $ of $ J_{\alpha_d}(z) $ satisfies the Euler-Rayleigh bounds $ \frac{1}{(d+1)^{d+1} A(\alpha_d)} < \Delta_1 < \frac{1}{(d+1)^{d+1} A(\alpha_d)} $, with $ \Delta_1 = \sum_{n \geq 1} j^{- (d+1)}_{\alpha_d,n} $, and higher-order bounds are derived for $ \Delta_2 $ and $ \Delta_3 $.
• The function $ x \mapsto \log J_{\alpha_d}(x) / \log \left(1 - x^{d+1}/j_{\alpha_d,1}^{d+1} \right) $ is strictly decreasing on $ (0, j_{\alpha_d,1}) $, yielding the inequality $ 1 < \Sigma_{\alpha_d}(x) < \frac{j_{\alpha_d,1}^{d+1}}{(d+1)^{d+1} A(\alpha_d)} $.
• The function $ q_{\alpha_d}(x) = \exp\left( \log \left( \frac{x^{S(\alpha_d)}}{(d+1)^{S(\alpha_d)} A(\alpha_d)} \right) - \frac{x}{(d+1)^{d+1} A(\alpha_d)} \right) $ is absolutely monotonic on $ [0, j_{\alpha_d,1}^{d+1}) $, leading to the Redheffer-type inequality $ J_{\alpha_d}(x^{1/(d+1)}) \leq \frac{x^{S(\alpha_d)/(d+1)}}{(d+1)^{S(\alpha_d)} A(\alpha_d)} \exp\left( - \frac{x}{(d+1)^{d+1} A(\alpha_d)} \right) $.