[Paper Review] The Modified Lommel functions: monotonic pattern and inequalities

Saiful R. Mondal|arXiv (Cornell University)|Apr 15, 2017

Mathematical Inequalities and Applications15 references3 citations

TL;DR

This paper investigates the monotonicity, log-convexity, and ratio properties of modified Lommel functions using their power series and infinite product representations. It establishes Turán-type and reverse Turán-type inequalities, derives a Rayleigh-type function for Lommel functions, and proves a Redheffer-type inequality, providing sharp bounds for these special functions in mathematical physics and analysis.

ABSTRACT

This article studies the monotonicity, log-convexity of the modified Lommel functions by using its power series and infinite product representation. Same properties for the ratio of the modified Lommel functions with the Lommel function, $\sinh$ and $\cosh$ are also discussed. As a consequence, some Turán type and reverse Turán type inequalities are given. A Rayleigh type function for the Lommel functions are derived and as an application, we obtain the Redheffer-type inequality.

Motivation & Objective

• To analyze the monotonicity and log-convexity properties of the normalized modified Lommel function $\lambda_{\mu,\nu}(x)$ with respect to parameters $\mu$, $\nu$, and variable $x$.
• To derive Turán-type and reverse Turán-type inequalities for the modified Lommel function and its ratios with $\sinh(x)$, $\cosh(x)$, and the Lommel function.
• To investigate the log-convexity of the ratio $L_{\mu-1/2,1/2}(x)/S_{\mu-1/2,1/2}(x)$ and related functions using infinite product factorizations.
• To construct a Rayleigh-type function for the Lommel function and apply it to derive a Redheffer-type inequality.
• To establish sharp inequalities and asymptotic bounds for the zeros of associated hypergeometric functions via series and product expansions.

Proposed method

• Utilizes the power series representation of $\lambda_{\mu,\nu}(x)$ as an even entire function of order one, expressed via the generalized hypergeometric function ${}_1F_2$.
• Applies the infinite product representation $\varphi_k(x) = \prod_{j=1}^\infty \left(1 - \frac{x^2}{\eta_{\mu,k,n}^2}\right)$ for associated functions to analyze zero distribution and log-convexity.
• Employs Lemma 1.1 on ratio monotonicity of power series with positive coefficients to prove monotonicity of $\lambda_{\mu,\nu}(x)/\lambda_{\mu_1,\nu_1}(x)$ and related ratios.
• Derives a Rayleigh-type function $\alpha_n^{(2m)} = \sum_{n=1}^\infty \eta_{\mu,n}^{-2m}$ from logarithmic differentiation of the infinite product for $\Lambda_{\mu-1/2,1/2}(x)$.
• Uses logarithmic differentiation and series re-expansion to relate the logarithmic derivative of $\Lambda_{\mu-1/2,1/2}(x)$ to the Rayleigh function and prove monotonicity of the ratio $\varphi_\mu(x)$.
• Applies the limit comparison principle and asymptotic analysis to evaluate boundary limits of ratios of derivatives, establishing strict monotonicity and convexity.

Experimental results

Research questions

• RQ1 Under what conditions on $\mu, \nu, \mu_1, \nu_1$ is the ratio $\lambda_{\mu,\nu}(x)/\lambda_{\mu_1,\nu_1}(x)$ strictly increasing on $(0, \infty)$?
• RQ2 How do the parameters $\mu$ and $\nu$ affect the log-convexity and monotonicity of $\lambda_{\mu,\nu}(x)$ for fixed $x > 0$?
• RQ3 What is the log-convexity behavior of the ratio $L_{\mu-1/2,1/2}(x)/S_{\mu-1/2,1/2}(x)$ on intervals between consecutive zeros of the Lommel function?
• RQ4 Can a Rayleigh-type function be constructed for the Lommel function to derive a Redheffer-type inequality?
• RQ5 What are the sharp bounds for the ratio $\lambda_{\mu,\nu}(x)/\cosh(x)$, and under what conditions is it strictly decreasing?

Key findings

• The ratio $\lambda_{\mu,\nu}(x)/\lambda_{\mu_1,\nu_1}(x)$ is strictly increasing on $(0, \infty)$ if $\mu_1 \geq \mu > -1$ and $(\mu_1 - \mu)(\mu_1 + \mu + 6) \geq \nu_1^2 - \nu^2$.
• For $\mu \pm \nu + 3 > 0$, the function $\mu \mapsto \lambda_{\mu,\nu}(x)$ is strictly decreasing and log-convex on $(-1, \infty)$ for each fixed $\nu \in \mathbb{R}$ and $x > 0$.
• The function $\nu \mapsto \lambda_{\mu,\nu}(x)$ is strictly log-convex on $\mathbb{R}$ for each fixed $\mu > -1$ and $x > 0$, provided $\mu \pm \nu + 3 > 0$.
• The function $x \mapsto \lambda_{\mu,\nu}^2(x)/\cosh(x)$ is strictly decreasing on $[0, \eta_{\mu,1})$ when $\mu \pm \nu + 3 > 0$.
• The ratio $\lambda_{\mu-1/2,1/2}(x)\Lambda_{\mu-1/2,1/2}(x)$ is strictly log-convex on $I_\mu$, the interval between the first two zeros of the Lommel function.
• The function $\varphi_\mu(x) = \log(\Lambda_{\mu-1/2,1/2}(x)) / \log(1 - x^2/\eta_{\mu,1}^2)$ is strictly decreasing on $[0, \eta_{\mu,1})$, with limits $\eta_{\mu,1}^2 \alpha_\mu^{(2)}$ at 0 and 1 at the endpoint, implying sharp bounds for the zero distribution.