Inequalities for modified Bessel functions and their integrals

Simple inequalities for some integrals involving the modified Bessel functions $I_{\nu}(x)$ and $K_{\nu}(x)$ are established. We also obtain a monotonicity result for $K_{\nu}(x)$ and a new lower bound, that involves gamma functions, for $K_0(x)$.

Whilst formula (1.1) holds for complex-valued z and ν, throughout this paper we shall restrict our attention to the case of real-valued z and ν. There are no closed form expressions in terms of modified Bessel and Struve functions in the literature for the integrals for the case β = 0. Moreover, even in the case β = 0 the expression on the right-hand side of formula (1.1) is a complicated expression involving the modified Struve function L ν (x). This provides the motivation for establishing simple bounds, in terms of modified Bessel functions, for the integrals defined in the first display.
In this paper we establish, through the use of elementary properties of modified Bessel functions and straightforward calculations, simple bounds, that involve modified Bessel functions, for the integrals given in the first display. Our bounds prove to be very useful when applied to calculations that arise in the study of Stein's method for Variance-Gamma distributions. We also obtain a monotonicity result and bound for the modified Bessel function of the second kind K ν (x), as well as a simple but remarkably tight lower bound for K 0 (x). These bounds are, again, motivated by the need for such bounds in the study of Stein's method for Variance-Gamma distributions. However, the bounds obtained in this paper may also prove to be useful in other problems related to modified Bessel functions; see for example, Baricz and Sun [3] in which inequalities for modified Bessel functions of the first kind were used to obtain lower and upper bounds for integrals of involving modified Bessel functions of the first kind. Throughout this paper we make use of some elementary properties of modified Bessel functions and these are stated in the appendix.

Inequalities for integrals involving modified Bessel functions
Before presenting our first result concerning inequalities for integrals of modified Bessel functions, we introduce some notation for the repeated integral of the function e βx x ν I ν (x), which will be used in the following theorem. We define (2.1) x 0 I (ν,β,n) (y) dy, n = 0, 1, 2, 3, . . . .
(ii) Using inequality (A.7) and then applying (A.12) we get (iii) From inequality (2.3), we have Integrating both sides of the above display n times with respect to x yields the desired inequality.
(iv) We prove the result by induction on n. The result is trivially true for n = 0. Suppose the result is true for n = k. From the inductive hypothesis we have Integration by parts and inequality (2.4) gives Rearranging we obtain and substituting into (2.8) gives Hence the result has been proved by induction.
Integrating both sides over (0, x), applying the fundamental theorem of calculus and rearranging gives The result now follows from the fact that I ν (x) > 0 for x > 0 and by the positivity of the integral.
(vi) From inequality (2.6) we have and Iterating gives the result.
We now state a simple lemma (which is a special case of Lemma 2.4 of Ismail and Muldoon [8]), that gives a monotonicity result for the ratio Kν−1(x) Kν (x) . The lemma has an immediate corollary, which we will make use of in the proof of our next theorem.
Proof. By the quotient rule and differentiation formula (A.14), we have d dx .
With the aid of Corollary 2.3 and standard properties of the modified Bessel function K ν (x), we can prove at the following theorem.
(ii) Using inequality (A.8) and then apply the differentiation formula (A.13) we have (iii) Now suppose that ν < 1/2 and β > 0. Using integration by parts and the differentiation formula (A.13) gives Applying the inequality (A.8) and rearranging gives Inequality (2.10) for β > 0 now follows on rearranging. The case β ≤ 0 is simple. Since e βt is a non increasing function of t when β ≤ 0 we have where we used inequality (2.9) to obtain the second inequality. Hence inequality (2.10) has been proved.
Now suppose ν > 1/2. We begin by defining the function u(x) to be We now show that u(x) ≥ 0 for all x ≥ 0, which will prove the result. We begin by noting that lim x→0 + u(x) = 0 and lim x→∞ u(x) = 0, which are verified by the following calculations, where we make use of the asymptotic formula (A.3) and the definite integral formula (A.17).
where we used the asymptotic formula (A.5) to obtain the above equality. We may obtain an expression for the first derivative of u(x) by the use of the differentiation formula (A.13) as follows In the limit x → 0 + we have, by the asymptotic formula (A.3), that Since ν > |ν − 1| for ν > 1/2 and lim x→0 + x a log x = 0, where a > 0, we have u ′ (x) ∼ 2 ν−1 Γ(ν), as x → 0 + , for ν > 1/2.
Therefore u(x) is initially an increasing function of x. In the limit x → ∞ we have, by (A.5), We therefore see that u(x) is an decreasing function of x for large, positive x. From the formula (2.13) we see that x * is a turning point of u(x) if and only if (2.14) From Corollary 2.3, it follows that equation (2.14) has one root for ν > 1/2 (for which √ πΓ(ν+1/2) Γ(ν) > 1). Putting these results together, we see that u(x) is non-negative at the origin and initially increases until it reaches it maximum value at x * , it then decreases and tends to 0 as x → ∞. Therefore u(x) is non-negative for all x ≥ 0 when ν > 1/2.
We now show that v(x) ≥ 0 for all x ≥ 0, which will prove the result. We begin by noting that lim x→0 + v(x) > 0 and lim x→∞ v(x) = 0, which are verified by the following calculations, where we make use of the asymptotic formula (A.3) and the definite integral formula (A.17).
where we used the asymptotic formula (A.5) to obtain the above equality. We may obtain an expression for the first derivative of v(x) by the use of the differentiation formula (A.13) as follows In the limit x → 0 + we have, by the asymptotic formula ( As in part (iv), we see that v(x) is initially an increasing function of x. In the limit Now, for ν > 1/2 and 0 < β < 1 we have, by (A.5), Hence, v(x) is an decreasing function of x for large, positive x. From formula (2.15) we see that x * is a turning point of v(x) if and only if Inequality (2.16) shows that N > 1 + N β for all ν > 1/2 and 0 < β < 1. From Corollary 2.3, it follows that equation (2.17) has one root for positive x and therefore v(x) has one maximum which occurs at positive x. Putting these results together we see that v(x) is positive at the origin and initially increases until it reaches it maximum value at x * , it then decreases and tends to 0 as x → ∞. Therefore v(x) is non-negative for all x ≥ 0 when ν > 1/2, which completes the proof.
Combining the inequalities of Theorems 2.1 and 2.5 and the indefinite integral formula (1.1) we may obtain lower and upper bounds for the quantity L ν (x)L ν−1 (x)− L ν−1 (x)L ν (x). Here is an example: Corollary 2.6. Suppose ν > −1/2, then for all x > 0 we have Proof. From the asymptotic formulas (A.2) and (A.6) for I ν (x) and L(x), respectively, we have that Therefore, applying the indefinite integral formula (1.1) gives, for ν > −1/2, From inequalities (2.2) and (2.6) of Theorem 2.1, we have Substituting this inequality into (2.18) gives The desired inequality now follows from rearranging terms and an application of the standard formula xΓ(x) = Γ(x + 1).
Remark 2.7. The lower and upper bounds for I ν (x)L ν−1 (x) − I ν−1 (x)L ν (x) that are given in Corollary 2.6 are simple, but very tight for large ν.

Inequalities for the modified Bessel function of the second kind
We now present some simple inequalities for the modified Bessel function of the second kind K ν (x). The following theorem establishes an inequality for the modified Bessel function K ν (x) that is useful in the study of Stein's method for Variance-Gamma distributions (see Gaunt [5]). Theorem 3.1. Let ν > 0 and x ≥ 0, then is a monotone decreasing function of x on (0, ∞) and satisfies the following inequality , for x ≥ 0, ν > 1.
The lower bound is also valid for all ν > 0.
Proof. Applying the differentiation formula (A.14) gives d dx Using (A.11) we may simplify the numerator as follows Thus, proving that (3.1) is monotone decreasing reduces to proving that, for x > 0, is a monotone decreasing function of x and from the asymptotic formula (A.3) we see that its limit as x → 0 + is lim x→0 + (x ν+1 K ν−1 (x)+ 2x ν K ν (x)) = 2 · 2 ν−1 Γ(ν) = 2 ν Γ(ν). Therefore (3.4) is proved, and so (3.1) is monotone decreasing on (0, ∞). It is therefore bounded above and below its values in the limits x → ∞ and x → 0. These are calculated using the asymptotic formulas (A.5) and (A.4) and are given below: where the first limit holds for all ν > 0 and the second limit is valid for all ν > 1. This completes the proof.

as required
Remark 3.5. Luke [11] obtained the following bounds for K 0 (x): Numerical experiments show that the bounds of Luke and our lower bound of Corollary 3.3 are remarkably accurate for all but very small x, for which the logarithmic singularity of K 0 (x) blows up. The lower bound 8 √ x 8x+1 outperforms our bound lower bound of Γ(x+1/2) Γ(x+1) for x > 0.394 (3 d.p.), whilst our bound outperforms for x < 0.394 (3 d.p.), and performs considerably better for very small x.