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 Grun (J Math Anal Appl 408:91–101, 2013).