Sharp bounds for the ratio of modified Bessel functions
Fecha
2017-06-21Metadatos
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Let $I_{\nu }\left( x\right) $ be the modified Bessel functions of the first
kind of order $\nu $, and $S_{p,\nu }\left( x\right) =W_{\nu }\left(
x\right) ^{2}-2pW_{\nu }\left( x\right) -x^{2}$ with $W_{\nu }\left(
x\right) =xI_{\nu }\left( x\right) /I_{\nu +1}\left( x\right) $. We achieve
necessary and sufficient conditions for the inequality $S_{p,\nu }\left(
x\right) <u$ or $S_{p,\nu }\left( x\right) >l$ to hold for $x>0$ by
establishing the monotonicity of $S_{p,\nu }(x)$ in $x\in \left( 0,\infty
\right) $ with $\nu >-3/2$. In addition, the best parameters $p$ and $q$ are
obtained to the inequality $W_{\nu }\left( x\right) <\left( >\right) p+\sqrt{%
x^{2}+q^{2}}$ for $x>0$. Our main achievements improve some known results,
and it seems to answer an open problem recently posed by Hornik and Gr\"{u}n
in [13].