Abstract: The best bounds of the form B (alfa, beta,gamma, x) = (alfa +beta2+gamma2x2)/x for ratios of modified Bessel functions are characterized: if alfa, beta and gamma are chosen in such a way that B (alfa, beta, gamma, x) is a sharp approximation for phi nu (x) =Iv-1x/Iv(x) as x -- 0+ (respectively x -- + infinito) and the graphs of the functions B (alfa, beta, gamma, x) and phi nu (x) are tangent at some x =xasterisco >0, then B (alfa, beta, gamma, x) is an upper (respectively lower) bound for phi nu (x) for any positive x, and it is the best possible at xasterisco. The same is true for the ratio phi nu (x) =K nu+1(x)/ K nu (x) but interchanging lower and upper bounds (and with a slightly more restricted range for nu). Bounds with maximal accuracy at 0+and +infinitoare recovered in the limits xasterisco --0+ and xasterisco -- +infinito, and for these cases the coefficients have simple expressions. For the case of finite and positive xasterisco we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.

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