Sharp bounds for the ratio of modified Bessel functions

Abstract

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].