Amos-type bounds for modified Bessel function ratios.

We systematically investigate lower and upper bounds for the modified Bessel function ratio [Formula: see text] by functions of the form [Formula: see text] in case [Formula: see text] is positive for all [Formula: see text], or equivalently, where [Formula: see text] or [Formula: see text] is a negative integer. For [Formula: see text], we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If [Formula: see text], the minimal elements are tangent to [Formula: see text] in exactly one point [Formula: see text], and have [Formula: see text] as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if [Formula: see text] is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.

Introduction

Let be the modified Bessel function of order , and the (modified) Bessel function ratio . These ratios are of great importance in a variety of application areas, including statistics  [e.g., 7] and numerical analysis  [e.g., 1], either directly or through the fact that by the well-known recurrence relations for modified Bessel functions, from which by integration and taking limits,

For functions and defined on the positive reals, write iff for all , with defined analogously. If neither nor , we say that and are incomparable. Let be a family of functions on the positive reals and . We say that is the least element (minimum) of iff for all , and that is a minimal element of iff there is no for which , with the greatest element (maximum) and maximal elements of defined analogously.

Let where in what follows we always (without loss of generality) take . For , Eqs. (9), (11) and (16) in Amos [1] show that Such “Amos-type” bounds were re-established and extended in several publications (see Section  3 for details). These bounds are very attractive because they allow both for explicit inversion and integration. Thus, Amos-type bounds yield bounds (and approximations) also for and the antiderivate of (equivalently, and its logarithm).

Let be the set of all for which is a lower/upper Amos-type bound for , and write for the corresponding families of lower/upper Amos-type bounds for .

In this paper, we investigate the structure of and under the condition that , or equivalently, or a negative integer.

Preliminaries

Let and so that .

Using, e.g., Watson  [10, Formula 3.7.2], If , all coefficients in the numerator and denominator series are non-negative and eventually positive, and hence . If is a negative integer, the same is true; otherwise, which is negative if , and hence for all sufficiently small positive .

Using the asymptotic expansion of for large argument [10, e.g., Formula 7.23.2], one can show that for arbitrary  , see also Schou  [7, Eq. (6), assuming  .

As is increasing with , we have iff . Hence, when or is a negative integer and is a (strict) upper or lower bound for if and only if is a (strict) lower or upper bound for , respectively.

For ,

More generally, if is not a negative integer, If is a negative integer, vanishes for from 0 to , and hence As for we have and , we can combine the two expansions to obtain the lemma. □

If ,For arbitrary and ,

If , then for , whence Eq. (3) by adding .

As and thus  □

For arbitrary or are only possible when or , respectively. If , then or are only possible when or , respectively.

The first assertion is immediate by comparing the expansions of and for . If has a unique zero , and changes from to at . If , so upper and lower bounds necessarily must have . The second assertion now follows by comparing the values of and at . □

Let and . Then iff , and iff . Otherwise, if and and for and for .

Consider . Then and as as as . As if we have and hence iff , and iff . As (or >) iff (or <). Otherwise, i.e., iff and has a unique zero in , which can be determined as follows. Let so that and , and iff Taking squares, from which Then The numerator equals so that indeed . Similarly, so that with we indeed obtain for the unique solution of (and equivalently ) on . Clearly, for and for , so that for and for , and the proof is complete. □

Suppose the quadratic polynomial has two real zeros . Then iff .

Trivial, as . □

Previous work

Amos [1] gives the bounds (Eq. (16)) and (Eqs. (9) and (11)). Using Lemma 3 with and we see that the first lower bound is uniformly better (larger) than the second one, whereas again with Lemma 3, neither of the upper bounds and is uniformly better (smaller) than the other: in fact, with and , we get so that for and for .

Nåsell [5] gives rational bounds for , and notes (p. 8) that the Amos-type bounds and are valid for and , respectively. But trivially , so that the upper bound is in fact valid for .

Simpson and Spector [9, Theorem 2] show that As the quadratic function has zeros Lemma 4 implies that and hence Using Lemma 3, we see that this bound is uniformly better than the Amos-type bound . To compare with , note that Thus, using Lemma 3 with and , we get and and therefore for , and for .

Neuman [6, Proposition 5] shows that As the quadratic function has zeros Lemma 4 implies that for and . If and hence .

Yuan and Kalbfleisch [11, Eq. (A.5)] show that

Baricz and Neuman [2, Theorems 2.1 and 2.2] show that if and , then and that (the reference uses for ). The former extends the earlier result of Neuman  [6] when , in which case the bounds are not valid for all . As has zeros Lemma 4 yields that for , the latter is equivalent to , extending the previously established range for this bound.

Laforgia and Natalini [4, Theorem 1.1] show that (the condition that is not stated explicitly in the theorem, but given in Eq. (1.8) of the reference used in the proof). As the result is equivalent to which is weaker than the bound.

Segura [8, Theorem 3] shows that or equivalently, for . For , Segura [8, Eqs. (22) and (61)] also shows that for and , Clearly, the lower bound is equivalent to for , and the upper bound to for and , which is weaker than the upper bound .

Kokologiannaki [3, Theorem 2.1] shows that for , for and . As the lower bound again is equivalent to for . Write for the above upper bound and . is the larger root of the quadratic polynomial so by Lemma 4, for any function with for all we have . Consider , and write . Then iff which in turn is equivalent to Let so that iff , and the inequality becomes The coefficient of the linear term is 0, so that and for we have for . Thus, for all . We thus have the following.

For all and ,

Hence, the upper bound in Kokologiannaki  [3, Theorem 2.1] is strictly weaker than the bound .

The various results can be summarized as follows: the “best” (in the sense of not being uniformly weaker than other) Amos-type bounds for currently available are

Results

For ,and is the maximum of the family of lower Amos-type bounds for .

We already know that for . By Theorem 1, is only possible if and . If , Lemma 3 implies that . Otherwise, we trivially have . □

For is a closed convex set.

For fixed is continuous, linear in , and satisfies and hence and is thus convex. By Theorem 1, is only possible when , for which it is equivalent to . Hence, is the intersection of closed convex sets, and thus a closed convex set. □

Let As for , clearly for .

For ,with continuous, decreasing and concave.

For , we have iff and . Thus, as is continuous and increasing in , if is non-empty, it is the closed interval . By Lemma 3, for all , so must be decreasing as long as is non-empty. If is decreasing and non-negative and thus must have a finite limit . Taking limits in and implies that and . Thus, is non-empty. As , the first assertion follows. Finally, as is closed and convex, must be continuous and concave. □

Let . For . For .

We know that . By Theorem 1, is only possible if so that . If and , by Eq. (3) and comparison with Eq. (2) shows that is only possible if in fact , or equivalently, if . For , Lemma 3 implies that , so that indeed . □

Let

Let . Then

As shown in Simpson and Spector [9], satisfies the Riccati equation and clearly, . Hence, as , with so that whence the lemma. □

Let (so that equals for and otherwise), and for let (where the second expressions shows that is well-defined).

Let . Then is strictly concave with equals if and if , and is non-negative and decreasing.

The assertions about the values of at and are straightforward. If and there is nothing left to prove. Hence, take . The second derivative of is given by For and we have and , giving numerators −2 and . Hence is the sum of two strictly concave functions, and thus strictly concave. Clearly, with value −1 at . By strict concavity, the derivative of is decreasing, and hence less than −1 for , so that the derivative of is negative for and is decreasing. It remains to show that . If , this is immediate from . Otherwise, and , which is non-negative as . □

Let . Then for .

The proof will be based on the ideas of Simpson and Spector [9]. Suppose is sufficiently often continuously differentiable on with . Suppose that for all implies that there exists a suitable odd such that for and . Then for all , as otherwise for we would have for all sufficiently small and a suitable , which is impossible.

In our case, , where . If , we have and we already know for that . By Lemma 6, is decreasing and hence maximal for with value . Thus, for we have , or equivalently, .

Write . If , which is only possible if , we have and for all . If , we already know that . Otherwise, . If for some , Lemma 5 implies that , completing the proof for this case.

Hence, consider the case where . Solving has discriminant with the larger root of the first factor. Hence, the discriminant vanishes, and with we have , where .

If for some , Lemma 5 implies that . If , and the proof is complete. Otherwise, use Lemma 5 to write , where so that and If and . Differentiation gives and , so that if and , and the proof is complete. □

Let . Then the elements of are mutually incomparable.

By Lemma 6, is decreasing, whence the result by using Lemma 3. □

For andFor .

Let . For arbitrary , by Eq. (4) and comparison with Eq. (1) shows that is only possible if or equivalently, if . For , the upper bound is negative, so that is impossible for all and hence . For , the condition is equivalent to . By Theorem 7, and by Theorem 1, , so that and . □

Let and . Then there exists a unique positive at which is tangent to . The map is continuous and increasing on , with and .

Write . By Theorem 6, we can find such that . Using Lemma 3 and the fact that , we can find such that for all for and for . If , we have for all sufficiently small that for . By the above, the same holds true for and . Hence, for all sufficiently small, which contradicts the maximality of . Thus, there must be at least one such that , and clearly, the derivatives must agree at as otherwise could not be an upper bound for . Equivalently, must be tangent to at . By Lemma 5, this is the case iff solves , from which we infer that is uniquely determined and continuous as a function of . The limits for from the right and from the left are obvious. To show that is increasing, it suffices to show that it is injective. Hence, let and suppose that . Then with , the must have the same value and derivative at , so that and hence , which is impossible as is decreasing by Theorem 5. □

Let . Then are the minimal elements of the family of upper Amos-type bounds for , and

Let . By Theorem 10, there exists a unique so that and hence , proving the second assertion. Let and . If , Theorem 6 shows that . If and , Lemma 3 implies that , which is impossible as by Theorem 10, must be the only tangent to at . Thus we always have , and again by Lemma 3, there always exists such that for and for . As for and trivially provided that , the first assertion follows, and the proof is complete. □

Finally, let us consider the cases where is a negative integer. As readily seen from the series expansion, , and hence .

If is a positive integer,and is the minimum of the family of upper Amos-type bounds for .

As and has a pole at , the same must be true for upper bounds of , implying that necessarily . As we have iff , i.e.,  . From the characterization of for (Theorem 3), this is possible iff and , or equivalently, . □

If is a positive integer,and is the maximum of the family of lower Amos-type bounds for .

For lower bounds of , we must have by the usual arguments, and Theorem 1 implies that necessarily . On the other hand, we also know that , or equivalently, , and the proof is complete. □

Note that for , we already know by Theorem 3 that is the greatest lower bound for , and Theorem 6 yields that , so that is the least upper bound for with .

Summary and conclusions

In this paper, we systematically investigate lower and upper Amos-type bounds for on the positive reals when is positive, or equivalently, when or is a negative integer.

For , the set of all giving lower bounds has a simple explicit description, and is the maximum of the family of lower Amos-type bounds for (Theorem 3).

For , the set of all giving upper bounds is of the form , where and is continuous, decreasing and concave (Theorem 5), with and for (Theorem 6). If , and by Theorem 9, and the upper bounds in the family are tangent to in exactly one point (Theorem 10, taking and ), and the minimal elements of the family of upper Amos-type bounds for , with as their lower envelope (Theorem 11).

Thus, for , the pointwise maximum over all lower Amos-type bounds equals , and hence is always smaller than . On the other hand, for , the pointwise minimum over all upper Amos-type bounds equals  .

For and , Theorems 7 and 8 establish a family of explicitly computable, mutually incomparable upper bounds for with . For , these bounds are new. For and , and Theorem 7 extends the range of the bound given in Simpson and Spector [9] from to , and for dominates as the best previously available upper bound with (and hence first order exact as ).

Finally, for the cases where is a negative integer, Theorems 12 and 13 give explicit characterizations of and , and establish and as the least upper and greatest lower Amos-type bounds for , respectively.

For , the value of is not known; the results in this paper imply that . It is also not known whether in this case can be obtained as the lower envelope of all upper Amos-type bounds. For , this is certainly not the case (as is the uniformly smallest upper bound). Hence the range deserves further investigation.