Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Inequalities and monotonicity results for zeros of modified Bessel functions of purely imaginary order


Author: Andrea Laforgia
Journal: Quart. Appl. Math. 44 (1986), 91-96
MSC: Primary 33A40; Secondary 34C10
DOI: https://doi.org/10.1090/qam/840446
MathSciNet review: 840446
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Abstract: Let $ {x_k}\left( v \right)$ and $ {x'_k}\left( v \right)$ denote the $ k$ th positive zeros, in decreasing order, of the modified Bessel function $ {K_{iv}}\left( x \right)$ of purely imaginary order and of its derivative $ {K'_{iv}}\left( x \right) = {\textstyle{d \over {dx}}}{K_{iv}}\left( x \right)$, respectively. We show that for $ k = 1,2,...$ and $ 0 < v < \infty ,{x_k}\left( v \right) < {x'_k}\left( v \right) < v$ and $ {x_k}\left( v \right)/{x_{k + 1}}\left( v \right)$ and $ {x_k}\left( v \right) - {x_{k + 1}}\left( v \right)$ decrease as $ k$ increases. Some related results are mentioned for the zeros of $ {K'_{iv}}\left( x \right)$ and the chain of inequalities $ \left\vert {{K_{iv}}\left( {{{x'}_1}\left( v \right)} \right)} \right\vert > \... ...eft( {{{x'}_{n + 1}}\left( v \right)} \right)} \right\vert > \cdot \cdot \cdot $ is established.


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DOI: https://doi.org/10.1090/qam/840446
Article copyright: © Copyright 1986 American Mathematical Society

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