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On approximating the modified Bessel function of the second kind.

Zhen-Hang Yang1,2, Yu-Ming Chu1

  • 1School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000 China.

Journal of Inequalities and Applications
|March 3, 2017
PubMed
Summary
This summary is machine-generated.

This study establishes new double inequalities for the modified Bessel function of the second kind. These findings provide precise bounds and conditions for the function

Keywords:
gamma functionmodified Bessel functionmonotonicity

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Area of Science:

  • Mathematical Analysis
  • Special Functions
  • Bessel Functions

Background:

  • The modified Bessel function of the second kind (often denoted as Kν(x)) is a crucial function in various scientific and engineering fields.
  • Understanding the properties and inequalities related to Bessel functions is essential for accurate modeling and analysis.

Purpose of the Study:

  • To establish and prove new double inequalities involving the modified Bessel function of the second kind.
  • To derive bounds for related mathematical expressions using these inequalities.
  • To determine the conditions under which a specific function, derived from Bessel functions, is strictly increasing or decreasing.

Main Methods:

  • The study employs analytical methods to prove the double inequalities.
  • It involves deriving necessary and sufficient conditions for the inequalities to hold.
  • Applications are explored through the derivation of bounds and monotonicity conditions.

Main Results:

  • The core result is the proof of specific double inequalities for the modified Bessel function of the second kind, Kν(x).
  • These inequalities are shown to hold if and only if certain parameter conditions are met.
  • New bounds for the expression [Formula: see text] are provided for specific ranges of [Formula: see text].

Conclusions:

  • The established inequalities offer a refined understanding of the behavior of the modified Bessel function of the second kind.
  • The derived bounds and monotonicity conditions have practical implications in mathematical analysis and applied mathematics.
  • This work contributes to the theoretical framework of special functions and their applications.