Abstract
Some new inequalities for quotients of modified Bessel functions of the first and second
kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first
and second kinds, higher-order monotonicity properties of these functions and applications to a special
function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used
criterion for the monotonicity of the quotient of two Maclaurin series.
Results & Lemmas (5)
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Lemma 2.1.
Lemma 2.1. Consider the power series f(x) = a0 + a1x + · · · + anxn + · · · and g(x) = b0 + b1x + · · · + bnxn + · · ·, where, for all n…
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.
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…
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. https://doi.org/10.1017/S0013091508001016 Published online by Cambridge University Press
Theorem 1
Theorem 1] that ν →Iν(x) is decreasing on (−1, ∞) for each fixed x > 0. However, an alternative derivation of the above inequality was…
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.
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) =…
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
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,…
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), https://doi.org/10.1017/S0013091508001016 Published online by Cambridge University Press
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