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Thread: Bessel function inequality

  1. #1
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    Bessel function inequality

    Can anybody help me to prove
    $\displaystyle 2x(K_1(x) - K_0(x)) - K_0(x) < 0$
    where $\displaystyle x > 0$ and $\displaystyle K_i$ is the modified Bessel function of the second kind.

    Thanks in advance!
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  2. #2
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    Sep 2011
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    Re: Bessel function inequality

    I solved it myself by noting that $\displaystyle K_1(x) = -K'_0(x)$, isolating $\displaystyle K'_0(x)/K_0(x)$, and applying Lemma 1 in Palítsev (1999) "Two-Sided Bounds Uniform in the Real Argument and the Index for Modified Bessel Functions."
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