Sharp Bounds for the Ratio of Modified Bessel Functions

Abstract

Let $I_{\nu }( x)$ be the modified Bessel functions of the first kind of order $\nu$, and $S_{p,\nu }( x) =W_{\nu }( x) ^{2}-2pW_{\nu }( x) -x^{2}$ with $W_{\nu }( x) =xI_{\nu }( x) /I_{\nu +1}( x)$. We achieve necessary and sufficient conditions for the inequality $S_{p,\nu }( x) <u$ or $S_{p,\nu }( x) >l$ to hold for $x>0$ by establishing the monotonicity of $S_{p,\nu }(x)$ in $x\in ( 0,\infty )$ with $\nu >-3/2$. In addition, the best parameters p and q are obtained to the inequality $W_{\nu }( x) <( >) p+\sqrt{ x^{2}+q^{2}}$ for $x>0$. Our main achievements improve some known results, and it seems to answer an open problem recently posed by Hornik and Grün (J Math Anal Appl 408:91–101, 2013).