Bounds for ratios of modified Bessel functions and associated Turán-type inequalities

摘要

New sharp inequalities for the ratios of Bessel functions of consecutive orders are obtained using as main tool the first order difference-differential equations satisfied by these functions; many already known inequalities are also obtainable, and most of them can be either improved or the range of validity extended. It is shown how to generate iteratively upper and lower bounds, which are converging sequences in the case of the I-functions. Few iterations provide simple and effective upper and lower bounds for approximating the ratios Iν(x)/Iν-1;(x) and the condition numbers xI'ν(x)/Iν(x) for any x,ν≥0; for the ratios Kν(x)/Kν+1(x) the same is possible, but with some restrictions on ν. Using these bounds Turán-type inequalities are established, extending the range of validity of some known inequalities and obtaining new inequalities as well; for instance, it is shown that Kν+1(x)Kν-1(x)/(Kν(x))2<|ν|/(|ν|-1), x>0, ν∈[-1,1] and that the inequality is the best possible; this proves and improves an existing conjecture.