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Monotonicity of Three Classes of functions Involving modified bessel functions of the second kind

BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY 2023年第5期49卷 1-25页

作者： Mao, Zhong-Xuan Tian, Jing-Feng North China Elect Power Univ Dept Math & Phys Hebei Key Lab Phys & Energy Technol Yonghua St 619 Baoding 071003 Peoples R China

In this paper, we determine the monotonicity of three classes of functions involving modified bessel functions of the second kind. We also presented some inequalities and bounds for modified bessel functions of the se... 详细信息

关键词： modified bessel functions of the second kind Monotonicity Monotonicity rules Airy function

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THE MONOTONICITY AND CONVEXITY FOR THE RATIOS OF modified bessel functions of the second kind AND APPLICATIONS

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2017年第7期145卷 2943-2958页

作者： Yang, Zhen-Hang Zheng, Shen-Zhou Beijing Jiaotong Univ Dept Math Beijing 100044 Peoples R China State Grid Zhejiang Elect Power Co Res Inst Dept Sci & Technol Hangzhou 310014 Zhejiang Peoples R China

Let K-v (x) be the modified bessel functions of the second kind of order v. We prove that the function x bar right arrow K-u (x) K-v (x) / K(u+ v)/2 (x)(2) is strictly decreasing on (0,8). Our study not only involves the Turan type inequalities, log-convexity or log-concavity of K-v (x), and the conjecture posed by Baricz, but also yields various new results concerning the monotonicity and convexity of the ratios of the modified bessel functions of the second kind. As applications of our main theorems, some new sharp inequalities involving K-v (x) are presented, which contain sharp estimates for Kv (x) and sharp bounds for the ratios K-v' (x)/K-v (x) and Kv+1 (x) / K-v (x).

关键词： modified bessel functions of the second kind monotonicity convexity functional inequality Turan type inequality

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Asymptotic matching the self-consistent expansion to approximate the modified bessel functions of the second kind

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2024年第30期57卷 305002-305002页

作者： Steinbock, Chanania Katzav, Eytan Hebrew Univ Jerusalem Racah Inst Phys IL-9190401 Jerusalem Israel

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem-too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified bessel functions of the second kind, K alpha(x) .

关键词： self-consistent expansion modified bessel functions of the second kind asymptotic matching perturbative expansions partition functions special functions

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Monotonicity rules for the ratio of two Laplace transforms with applications

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2019年第2期470卷 821-845页

作者： Yang, Zhenhang Tian, Jing-Feng North China Elect Power Univ Coll Sci & Technol Baoding 071051 Hebei Peoples R China

Let f and g be both continuous functions on (0, infinity) with g (t) > 0 for t is an element of (0, infinity) and let F (x) = L (f), G (x) = L (g) be respectively the Laplace transforms of f and g converging for x > 0. We prove that if there is a t* E (0, infinity) such that f/g is strictly increasing on (0, t*) and strictly decreasing on (t*, infinity), then the ratio F/G is decreasing on (0, infinity) if and only if H-F,H-G (0(+)) - lim(x -> 0+) (F'(x)/G'(x)G(x) - F(x)) >= 0, with lim(x -> 0+) F(x)/G(x) = lim(t ->infinity) f(t)/g(t) and lim(x ->infinity) F(x)/G(x) = lim(t ->infinity) f(t)/g(t) provided the indicated limits exist. While H-F,H-G (0(+)) < 0, there is at least one x* > 0 such that F/G is increasing on (0, x*) and decreasing on (x*, infinity). As applications, a unified treatment for certain bounds of psi function is presented, and some properties of the modified bessel functions of the second are established. These show that the monotonicity rules in this paper may contribute to study for certain special functions because many special functions can be expressed as corresponding Laplace transforms. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Laplace transform Monotonicity rule Psi function modified bessel functions of the second kind

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The signs rule for the Laplace integrals with applications

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INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2024年第4期55卷 1416-1428页

作者： Yang, Zhen-Hang Tian, Jing-Feng State Grid Zhejiang Elect Power Co Res Inst Hangzhou 310014 Peoples R China North China Elect Power Univ Dept Math & Phys Hebei Key Lab Phys & Energy Technol Yonghua St 619 Baoding 071003 Peoples R China

Let the function f (t) have a unique zero on (0, 8). Then the signs of F (x) = ?(8)(0) f (t) e(-xt)dt on (0, 8) depends on the sign of F (0(+)). As applications, a double inequality involving the modified bessel functions of the second kind is extended, and a new proof of Alzer's inequalities for the gamma function is presented. Finally, we introduce a notion of incompletely monotonic functions of order n, which reduces to the usual notion of completely monotonic functions when the order is infinite.

关键词： Laplace integral Signs rule modified bessel functions of the second kind Gamma function Inequality

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Saddlepoint approximation for the generalized inverse Gaussian Levy process

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2022年第0期411卷

作者： Zhang, Mimi Revie, Matthew Quigley, John Trinity Coll Dublin Sch Comp Sci & Stat Dublin Ireland Univ Strathclyde Dept Management Sci Glasgow Scotland Sci Fdn Ireland I Form Adv Mfg Res Ctr Dublin Ireland

The generalized inverse Gaussian (GIG) Levy process is a limit of compound Poisson processes, including the stationary gamma process and the stationary inverse Gaussian process as special cases. However, fitting the GIG Levy process to data is computationally intractable due to the fact that the marginal distribution of the GIG Levy process is not convolution-closed. The current work reveals that the marginal distribution of the GIG Levy process admits a simple yet extremely accurate saddlepoint approximation. Particularly, we prove that if the order parameter of the GIG distribution is greater than or equal to -1, the marginal distribution can be approximated accurately - no need to normalize the saddlepoint density. Accordingly, maximum likelihood estimation is simple and quick, random number generation from the marginal distribution is straight-forward by using Monte Carlo methods, and goodness-of-fit testing is undemanding to perform. Therefore, major numerical impediments to the application of the GIG Levy process are removed. We demonstrate the accuracy of the saddlepoint approximation via various experimental setups. (C)& nbsp;2022 The Author(s). Published by Elsevier B.V.

关键词： Metropolis-Hastings algorithm modified bessel functions of the second kind Parametric bootstrap Saddlepoint approximation

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Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

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MATERIALS 2023年第9期16卷 3565-3565页

作者： Stankiewicz, Anna Univ Life Sci Lublin Fac Prod Engn Dept Technol Fundamentals PL-20612 Lublin Poland

The viscoelastic relaxation spectrum is vital for constitutive models and for insight into the mechanical properties of materials, since, from the relaxation spectrum, other material functions used to describe rheological properties can be uniquely determined. The spectrum is not directly accessible via measurement and must be recovered from relaxation stress or oscillatory shear data. This paper deals with the problem of the recovery of the relaxation time spectrum of linear viscoelastic material from discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation test. A two-level identification scheme is proposed. In the lower level, the regularized least-square identification combined with generalized cross-validation is used to find the optimal model with an arbitrary time-scale factor. Next, in the upper level, the optimal time-scale factor is determined to provide the best fit of the relaxation modulus to experiment data. The relaxation time spectrum is approximated by a finite series of power-exponential basis functions. The related model of the relaxation modulus is proved to be given by compact analytical formulas as the products of power of time and the modified bessel functions of the second kind. The proposed approach merges the technique of an expansion of a function into a series of independent basis functions with the least-squares regularized identification and the optimal choice of the time-scale factor. Optimality conditions, approximation error, convergence, noise robustness and model smoothness are studied analytically. Applicability ranges are numerically examined. These studies have proved that using a developed model and algorithm, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and noise robust;small model errors are obtained for the optimal time-scale factors. The complete scheme of the hierarchical computations is outlined, whic

关键词： viscoelasticity relaxation time spectrum linear relaxation modulus hierarchical identification algorithm regularized least-squares identification time-scale factor optimal selection modified bessel functions of the second kind singular value decomposition

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