含参数 Nabla 积分比和变限含参数 Nabla 积分比的单调性法则及其应用

摘要/Abstract

摘要：

利用时标理论中的 Nabla 积分建立了含参数 Nabla 积分比

$s\mapsto \frac{\int_\alpha^\beta \Psi(s,v) \nabla v}{\int_\alpha^\beta \Phi(s,v) \nabla v} \quad \text{和} \quad s\mapsto \frac{\int_{v_0}^\infty \Psi(s,v) \nabla v}{\int_{v_0}^\infty \Phi(s,v) \nabla v}$

以及变限含参数 Nabla 积分比

$s\mapsto \frac{\int_{s}^\infty \psi(v) w(s,v) \nabla v }{\int_{s}^\infty \phi(v) w(s,v) \nabla v} \quad \text{和} \quad s\mapsto \frac{\int_{v_0}^{s} \psi(v) w(s,v) \nabla v }{\int_{v_0}^{s} \phi(v) w(s,v) \nabla v}$

的单调性法则. 在含参数 Nabla 积分比部分中, 还详细研究了一些特殊情形, 包括时标下的多项式之比以及 Nabla 拉普拉斯变换之比. 利用这些单调性法则, 证明了函数 $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}_{u_i}(s)}{n \mathcal{J}_{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}_{v}(u_is)}{n \mathcal{J}_{v}(\bar{u}s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n K_{u_i}(s)}{n K_{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}_{u_i}(s)}{n \mathcal{Y}_{\bar{u}}(s)} $ 和 $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}_{v}(u_is)}{n \mathcal{Y}_{v}(\bar{u}s)} $ 的单调性, 其中 $ \bar{u}=\sum\limits_{i=1}^n u_i/n $, $ I_u(\cdot), K_u(\cdot) $ 分别为第一类和第二类修正的贝塞尔函数, $ \mathcal{J}_u(s):= \big( \frac{s}{2} \big)^{-u} I_{u}(s) $ 和 $ \mathcal{Y}_u(s):=K_u(s)-K_0(s) $.

关键词: 单调性法则, 时标, Nabla 积分, 修正贝塞尔函数, 拉普拉斯变换

Abstract:

Using the Nabla integral on time scales, this paper establishes the monotonicity rules for the ratios of parametric Nabla integrals

$s\mapsto \frac{\int_\alpha^\beta \Psi(s,v) \nabla v}{\int_\alpha^\beta \Phi(s,v) \nabla v} \quad \text{and} \quad s\mapsto \frac{\int_{v_0}^\infty \Psi(s,v) \nabla v}{\int_{v_0}^\infty \Phi(s,v) \nabla v}$

and the ratios of the parametric Nabla integrals with variable limits

$s\mapsto \frac{\int_{s}^\infty \psi(v) w(s,v) \nabla v }{\int_{s}^\infty \phi(v) w(s,v) \nabla v} \quad \text{and} \quad s\mapsto \frac{\int_{v_0}^{s} \psi(v) w(s,v) \nabla v }{\int_{v_0}^{s} \phi(v) w(s,v) \nabla v}.$

In the part of monotonicity rules for the ratios of parametric Nabla integrals, some different special cases are considered in detail, including the ratio of two polynomials on time scales and the ratio of two Nabla Laplace transforms. Using these monotonicity rules, the monotonicity of the functions $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}{u_i}(s)}{n \mathcal{J}{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}{v}(u_is)}{n \mathcal{J}{v}(\bar{u}s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n K_{u_i}(s)}{n K_{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}{u_i}(s)}{n \mathcal{Y}{\bar{u}}(s)} $ and $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}{v}(u_is)}{n \mathcal{Y}{v}(\bar{u}s)} $ is proved, where $ \bar{u}=\sum\limits_{i=1}^n u_i/n $, $ I_u(\cdot), K_u(\cdot) $ are the modified Bessel functions of the first and second kind, respectively, $ \mathcal{J}_u(s):= \big( \frac{s}{2} \big)^{-u} I_{u}(s) $ and $ \mathcal{Y}_u(s):=K_u(s)-K_0(s) $.

Key words: Monotonicity rules, Time scales, Nabla integral, Modified Bessel functions, Laplace transforms

中图分类号:

O171

田景峰, 毛忠旋, 孙龙发. 含参数 Nabla 积分比和变限含参数 Nabla 积分比的单调性法则及其应用[J]. 数学物理学报, 2024, 44(2): 298-312.

Tian Jingfeng, Mao Zhongxuan, Sun Longfa. Monotonicity Rules of the Ratios of Parametric Nabla Integrals and Parametric Nabla Integrals with Variable Limits and Their Applications[J]. Acta mathematica scientia,Series A, 2024, 44(2): 298-312.

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