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Amos-type bounds for modified Bessel function ratios.

Kurt Hornik1, Bettina Grün2

  • 1Institute for Statistics and Mathematics, WU Wirtschaftsuniversität Wien, Augasse 2-6, 1090 Vienna, Austria.

Journal of Mathematical Analysis and Applications
|June 14, 2014
PubMed
Summary
This summary is machine-generated.

This study establishes bounds for modified Bessel function ratios using specific function forms. Researchers identified greatest lower bounds and characterized minimal upper bounds for various conditions, aiding mathematical analysis.

Keywords:
BoundsInequalitiesModified Bessel function ratioModified Bessel functions of the first kind

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Area of Science:

  • Mathematical Analysis
  • Special Functions Theory

Background:

  • Modified Bessel functions are crucial in various scientific and engineering fields.
  • Understanding the behavior of ratios of these functions is essential for accurate modeling and analysis.

Purpose of the Study:

  • To systematically investigate and establish lower and upper bounds for the modified Bessel function ratio Iν(x)/Iν+1(x).
  • To analyze these bounds for different conditions of the parameter ν, including positive real numbers and negative integers.

Main Methods:

  • Utilized functions of the form x^α * exp(βx) to derive bounds.
  • Analyzed the properties of the set of lower and upper bounds, including greatest and minimal elements.
  • Investigated tangency properties of minimal upper bounds for specific parameter values.

Main Results:

  • Provided an explicit description of the set of lower bounds for ν > 0, identifying a greatest element.
  • Characterized the set of upper bounds and its minimal elements, showing tangency to the function at a specific point.
  • Developed a new family of explicitly computable upper bounds and described bounds for negative integer values of ν.

Conclusions:

  • The study offers a comprehensive analysis of bounds for modified Bessel function ratios.
  • The findings provide valuable tools for approximating and analyzing these functions in mathematical and applied contexts.