双参数广义 Nielsen 贝塔函数的不等式与渐近逼近

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 575-585.

双参数广义 Nielsen 贝塔函数的不等式与渐近逼近

尹枥^*(),黄利国(),张聚梅()

滨州学院理学院山东滨州 256603

收稿日期:2023-04-24 修回日期:2023-08-01 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *尹枥, Email：yinli_79@163.com
作者简介:黄利国, Email：liguoh123@bzu.edu.cn;|张聚梅, Email：wdzjm02@163.com
基金资助:
滨州学院重大项目(2020ZD02)

Inequality and Asymptotic Approximation Generalized Nielsen Beta Function with Two Parameters

Yin Li^*(),Huang Liguo(),Zhang Jumei()

College of Science, Binzhou University, Shandong Binzhou 256603

Received:2023-04-24 Revised:2023-08-01 Online:2024-06-26 Published:2024-05-17
Supported by:
Major Project of Binzhou University(2020ZD02)

摘要/Abstract

摘要：

该文介绍了经典 Nielsen 贝塔函数的一种双参数推广, 深入探讨了它的单调性、凹凸性、完全单调性与积分渐近性质, 并在此基础上建立了一些新的不等式.

关键词: Nielsen贝塔函数, 广义伽马函数, 渐近, 完全单调性, 不等式

Abstract:

In this paper, we introduces a two parameters generalization of the classical Nielsen beta function, discusses its monotonicity, concavity and convexity, complete monotonicity, and asymptotic properties of integrals, and establishes some new inequalities.

Key words: Nielsen beta function, Generalized gamma function, Asymptotic property, Complete monotonicity, Inequalities

中图分类号:

O178

尹枥, 黄利国, 张聚梅. 双参数广义 Nielsen 贝塔函数的不等式与渐近逼近[J]. 数学物理学报, 2024, 44(3): 575-585.

Yin Li, Huang Liguo, Zhang Jumei. Inequality and Asymptotic Approximation Generalized Nielsen Beta Function with Two Parameters[J]. Acta mathematica scientia,Series A, 2024, 44(3): 575-585.

参考文献

[1]	Baricz á, Bhayo B A, Pogány T K. Convexity properties of generalized trigonometric and hyperbolic functions. J Math Anal Appl, 2014, 417: 244-259
[2]	Boyadzhiev K N, Medina L A, Moll V H. The integral in Gradshteyr and Ryzhik, Part 11: The incomplete beta function. Scientia Ser A: Math Sci, 2009, 18(1): 61-75
[3]	Chen C P, Choi J S. Completely monotonic functions related to Gurland’s ratio for the gamma function. Math Inequal Appl, 2017, 20(3): 651-659
[4]	Copson E T. Asymptotic Expansions, Cambridge Tracts in Math Math Phys 55. New York:Cambridge University Press, 1965
[5]	Dubourdieu J, Sur un théorème de M S. Bernstein relatifàla transformation de Laplace-Stieltjes. Comp Math, 1939, 7: 96-111
[6]	Krasniqi V, Qi F. Complete monotonicity of a function involving the $p$-psi function and alternative proofs. Global Journal of Mathematical Analysis, 2014, 2(3): 204-208
[7]	Nantomah K. Convexity properties and inequalities concerning the $(p,k)$-gamma function. Commun Fac Sci Univ Ank Series A1, 2017, 66(2): 130-140
[8]	Nantomah K. Monotonicity and convexity properties and some inequalities involving a generalized form of the Wallis’ cosine formula. Research Journal of Math, 2017, 6(3): 1-10
[9]	Nantomah K. Monotonicity and convexity properties of the Nielsen’s $\beta$-function. Probl Anal Issues Anal, 2017, 24(2): 81-93
[10]	Nantomah K. On some properties and inequalities for the Nielsen’s $\beta$-function. Scientia Series A: Mathematical Sciences, 2017- 2018, 28: 43-54
[11]	Nantomah K, Merovci F, Nasiru S. Some inequalities and monotonicity properties for the $(p,k)$-gamma function. Kragujevac Journal of Math, 2018, 42(2): 287-297
[12]	Nantomah K, Prempeh E, Twum S B. On a $(p,k)$-analogue of the gamma function and some associated inequalities. Moroccan Journal of Pure and Applied Analysis, 2016, 2(2): 79-90
[13]	Nielsen N. Handbuch der Theorie der Gamma Function (First edition). Leipzig: B G Teubner, 1906
[14]	Qi F, Chen C P. Some completely monotonic and polygamma functions. J Aust Math Soc, 2006, 80: 81-88
[15]	Qi F, Guo B N. A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J Korean Math Soc, 2011, 48(3): 655-667
[16]	Qi F, Guo B N. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Commun Pure Appl Anal, 2009, 8(6): 1975-1989
[17]	Qi F, Guo B N. Necessary and sufficient conditons for functions involving the tri- and tetra-gamma functions to be completely monotonic. Adv Appl Math, 2010, 44(1): 71-83
[18]	Qi F, Guo S L, Guo B N. Completely monotonicity of some functions involving polygamma functions. J Comput Appl Math, 2010, 233: 2149-2160
[19]	Qi F, Huang Z. Inequalities of the complete elliptic integrals. Tamkang J Math, 1998, 29(3): 165-169
[20]	Shrafi K, Leon-Garcia A. Estimation of shape parameter for generalized Gaussian distribution in subband decomposition of video. IEEE Trans Circ Syst Video Techn, 1995, 5: 52-56
[21]	Stacy E W. A generalization of gamma distribution. Ann Math Statistics, 1962, 28: 1187-1192
[22]	Wang M K, Chu Y M. Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math Sci, 2017, 37B(3): 607-622
[23]	Wang M K, Chu Y M, Jiang Y P. Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mountain J Math, 2016, 46(2): 679-691
[24]	王淼坤, 褚玉明, 蒋月评, 闫丹丹. 零平衡超几何函数的一类二次变换不等式. 数学物理学报, 2014, 34A(4):999-1007
[24]	Wang M K, Chu Y M, Jiang Y P, Yan D D. A class of quadratic transformation inequalities for zero-balanced hypergeometric functions. Acta Math Sci, 2014, 34A(4):999-1007
[25]	Wang M K, Chu Y M, Zhang W. Monotonicity and inequalities involving zero-balanced hypergeometric function. Math Inequal Appl, 2019, 22(2): 601-617
[26]	王淼坤, 宋迎清, 褚玉明. 零平衡超几何函数的不等式. 数学学报, 2014, 57(4):719-726
[26]	Wang M K, Song Y Q, Chu Y M. A class of quadratic transformation inequalities for zero-balanced hypergeometric functions. Acta Math Sinica, 2014, 57(4):719-726
[27]	Wang M K, Zhang W, Chu Y M. Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math Sci, 2019, 39B(5): 1440-1450
[28]	Widder D V. The Laplace Transform. Princeton: Princeton University Press, 1941
[29]	Zhao T H, Wang M K, Zhang W, Chu Y M. Quadratic transformation inequalities for Gaussian hypergeometric function. J Inequal Appl, 2018, 2018: 1-15