Bounds for modified Bessel functions of the first and second kinds

Lemma 2.1. Consider the power series f(x) = a0 + a1x + · · · + anxn + · · · and g(x) = b0 + b1x + · · · + bnxn + · · · , where, for all n ⩾0, integer an ∈R and bn > 0, and suppose that both converge on R. If the sequence {an/bn}n⩾0 is (strictly) increasing (decreasing), then the function x →f(x)/g(x) is also (strictly) increasing (decreasing) on (0, ∞). For different proofs and various applications of this result the interested reader, is referred to [3,5,7–10,12,40,60,65] and the references the

Theorem 2.2. Let ν, µ > −1 and let k be a natural number. The following assertions are then true. (i) If ν < µ (ν > µ), then the function x →Iν(x)/Iµ(x) is strictly increasing (decreasing) on (0, ∞). (ii) The function x →I(2k) ν (x)/cosh x is strictly increasing on (0, ∞) for all ν < −1 2 and strictly decreasing on (0, ∞) for all ν > −1 2.

Theorem 1] that ν →Iν(x) is decreasing on (−1, ∞) for each fixed x > 0. However, an alternative derivation of the above inequality was given in 1972 by Luke [56, p. 63], by using a completely different approach. Clearly, Theorem 2.2 (iv) can be rewritten as follows: cosh x cosh y x y ν < Iν(x) Iν(y), (2.19) where 0 < x < y and ν > −1 2. Moreover, the above inequality is reversed when ν ∈ (−1, −1

Theorem 2.2. We note that it is easy to see that, from the inequality (2.14), by using the recurrence formula [76, p. 79] I′ ν(x)/Iν(x) = ν/x + Iν+1(x)/Iν(x), after integration we obtain the inequality (2.19). Analogously, from the inequality (2.16), after integration we get the inequality (2.20). Taking into account the proof of Theorem 2.2 (vi), these in turn imply that in fact the inequality (2.14) is equivalent to (2.19), while the inequality (2.16) is equivalent to (2.20). Finally, we propo

Theorem 2.2 (i) we obtain that, for all µ > ν > −1 and x > 0, the Tur´an-type inequality Iν+1(x)Iµ(x) −Iµ+1(x)Iν(x) > 0 holds. Clearly, this is equivalent to wµ(x) > wν(x),