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).
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This work was completed with the support of the Natural Science Foundation of China Grant no. 11371050 and NSFC-ERC Grant no. 11611530539.
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Yang, ZH., Zheng, SZ. Sharp Bounds for the Ratio of Modified Bessel Functions. Mediterr. J. Math. 14, 169 (2017). https://doi.org/10.1007/s00009-017-0971-1
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DOI: https://doi.org/10.1007/s00009-017-0971-1