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The monotonicity and convexity for the ratios of modified Bessel functions of the second kind and applications


Authors: Zhen-Hang Yang and Shen-Zhou Zheng
Journal: Proc. Amer. Math. Soc. 145 (2017), 2943-2958
MSC (2010): Primary 33C10, 26A48; Secondary 39B62, 26A51
DOI: https://doi.org/10.1090/proc/13522
Published electronically: January 23, 2017
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Abstract: Let $ K_{v}\left ( x\right ) $ be the modified Bessel functions of the second kind of order $ v$. We prove that the function $ x\mapsto K_{u}\left ( x\right ) K_{v}\left ( x\right ) /K_{\left ( u+v\right ) /2}\left ( x\right ) ^{2}$ is strictly decreasing on $ \left ( 0,\infty \right ) $. Our study not only involves the Turán type inequalities, log-convexity or log-concavity of $ K_{v}\left ( x\right ) $, 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}\left ( x\right ) $ are presented, which contain sharp estimates for $ K_{v}\left ( x\right ) $ and sharp bounds for the ratios $ K_{v}^{\prime }\left ( x\right ) /K_{v}\left ( x\right ) $ and $ K_{v+1}\left ( x\right ) /K_{v}\left ( x\right ) $.


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Additional Information

Zhen-Hang Yang
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Address at time of publication: Department of Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, People’s Republic of China, 310014
Email: yzhkm@163.com

Shen-Zhou Zheng
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Email: shzhzheng@bjtu.edu.cn

DOI: https://doi.org/10.1090/proc/13522
Keywords: Modified Bessel functions of the second kind, monotonicity, convexity, functional inequality, Tur\'an type inequality
Received by editor(s): June 23, 2016
Received by editor(s) in revised form: July 7, 2016, and August 4, 2016
Published electronically: January 23, 2017
Additional Notes: The second author was supported in part by the National Natural Science Foundation of China Grant #11371050.
Dedicated: This paper is dedicated to the 60th anniversary of Zhejiang Normal University
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society