Monotonicity of the ratio of modified Bessel functions of the first kind with applications.

Let [Formula: see text] with [Formula: see text] be the modified Bessel functions of the first kind of order v. In this paper, we prove the monotonicity of the function [Formula: see text] on [Formula: see text] for different values of parameter p with [Formula: see text]. As applications, we deduce some new Simpson-Spector-type inequalities for [Formula: see text] and derive a new type of bounds [Formula: see text] ([Formula: see text]) for [Formula: see text]. In particular, we show that the upper bound [Formula: see text] for [Formula: see text] is the minimum over all upper bounds [Formula: see text], where [Formula: see text] and is not comparable with other sharpest upper bounds. We also find such type of upper bounds for [Formula: see text] with [Formula: see text] and for [Formula: see text] with [Formula: see text].

Introduction

The modified Bessel functions of the first kind of order v, denoted by , are a class of particular solutions of the second-order differential equation [1, p. 77] which is represented explicitly by the infinite series where with  .

It is well known that the ratio plays an important role in the finite elasticity [2, 3] and epidemiological models [4, 5]. It was proved in [2, Theorem 2] by Simpson and Spector that is strictly increasing and convex on for , and the inequality holds for and . For this, such an inequality similar to (1.3) was called Simpson–Spector-type inequality for by Yang and Zheng [6, p. 2]. In [7, Proposition 5] Neuman presented a reversed version of (1.3): for and . In 2007, Baricz and Neuman [8, Theorem 2.2] extended the range of v from to such that is strictly increasing on , and showed that the inequality holds for and . Very recently, Yang and Zheng in [6] got the necessary and sufficient conditions for the Simpson–Spector-type inequality or to hold for by establishing the monotonicity of in with , where which actually answered an open problem recently posed by Hornik and Grün in [9]. Other similar or equivalent inequalities involving the ratio can be found in [10, Eqs. (11) and (16)], [11], [12, E1. (A.5)], [13], [14, Theorem 1.1], [15, Eqs. (22) and (61)], [9, 16–18] and the references therein.

Motivated by these above-mentioned recent papers, the main aim of this present paper is to prove the monotonicity of the function on for . Our main result is stated as follows.

Theorem 1.1

For , let the function be defined in by (1.6) and be defined by

If for or for , then the function is increasing from onto .

If , then is decreasing from onto .

If , then there exists an such that is increasing on , and decreasing on . Consequently, it holds that for , where , and is a unique solution of the equation on .

If for , then we have for , where . The lower and upper bounds for are sharp.

The rest of this paper is organized as follows. In Sect. 2, some lemmas are listed. The proof of Theorem 1.1 is presented in Sect. 3. In Sect. 4, as applications of Theorem 1.1, some Simpson–Spector-type inequalities for are established in Sect. 4.1; in Sect. 4.2, a new type of bounds () for for with is established, and a new Amos-type upper bound for is presented; some computable bounds for for with and for with are found in Sect. 4.3.

Lemmas

To prove Theorem 1.1, we need some lemmas. The following lemma which comes from [19, (3.5)] (see also [20]) is useful.

Lemma 2.1

Let be the modified Bessel function of the first kind of order v, which is showed by (1.2). Then we have In particular, we have

Lemma 2.2

([21])

Let and be two real power series converging on for some with for all k. If the sequence is increasing (or decreasing) for all k, then the function is also increasing (or decreasing) on .

Lemma 2.2 is a powerful tool to deal with the monotonicity of the ratio between two power series. An improvement of Lemma 2.2 has been presented in [22, Theorem 2.1]. A similar monotonicity rule for the ratio of two Laplace transforms was established in [23, Lemma 4] (see also [24]).

Lemma 2.3

Let and be two real power series converging on with for all k. If, for certain , the non-constant sequence is increasing (or decreasing) for and decreasing (or increasing) for , then there is a unique such that the function is increasing (or decreasing) on and decreasing (or increasing) on .

Lemma 2.3 first appeared in [25, Lemma 6.4] without giving the details of the proof. Two strict proofs were given in [22] and [26]. Another useful tool associated with Lemma 2.3 is the sign rule of a class of special series or polynomials, see, for example, [25, Lemma 6.3], [27, Lemma 7], [28]).

Lemma 2.4

([29, Problems 85, 94])

If two given sequences and satisfy the following conditions: then must be convergent for all values of t too, and

Proof of Theorem 1.1

Now we are in a position to prove Theorem 1.1.

Proof

Let us write as follows: where Using formulas (2.1) and (2.2), we have where and where Therefore, can be written in the form of A direct computation yields and from Lemma 2.4 we get

Therefore, to show the monotonicity of the ratio , it suffices to observe the monotonicity of the sequence . Since for and , we have where This shows that the sequence is increasing if , and increasing for then decreasing for if . Consequently, we deduce that for , if and if , where is given by (1.7).

Now we discuss the monotonicity of by dividing it into two cases.

Case 1. .

Subcase 1.1. . From relation (3.3) it is obtained that the sequence is increasing. By Lemma 2.2 it follows that the ratio is increasing on .

Subcase 1.2. . It is seen that the sequence is decreasing, and from Lemma 2.2 it follows that the ratio is decreasing on .

Subcase 1.3. . Noting that the sequence is decreasing, and it is seen that there exists such that for , and for . This implies that is increasing for and decreasing for . By Lemma 2.3 it is derived that there is such that the ratio is increasing on and decreasing on . Consequently, we have for , that is, inequalities (1.8) hold.

Case 2. .

Subcase 2.1. . In the same way, we get that the ratio is increasing on .

Subcase 2.2. . Similarly, we find that the ratio is decreasing on .

Subcase 2.3. . We have and notice that for . Hence, we get that for , This shows that the sequence is increasing for and decreasing for . By Lemma 2.3 it is derived that there is such that the ratio is increasing on and decreasing on . Therefore, inequality (3.7) holds, which implies inequalities (1.8).

Subcase 2.4. . We easily check that , and for , This yields for . Since , we see that the inequality is sharp.

The continuity of the function on together with and means that is bounded on , so exists, which completes the proof. □

Remark 3.1

In Subcase 2.4: for , we see that the sequence is increasing, and which implies that there exists such that for and for . This indicates that the sequence is decreasing for and increasing for . Since we find that the sequence is increasing for , decreasing for , and increasing for .

Clearly, we are not able to describe the monotone pattern of by directly using Lemmas 2.2 and 2.3. We here guess that there are two , with such that is increasing on and decreasing on .

Some new type of bounds for

Simpson–Spector-type inequality for

It is clear that where the latter indeed offers some new Simpson–Spector-type inequalities for . In fact, by Theorem 1.1 we immediately get the following.

Proposition 4.1

Let where is given in (1.7). Then the double inequality holds for and if and only if where if , and and here is the unique solution of the equation on with .

Proof

(i) By Theorem 1.1 we see that the left-hand side inequality of (4.1) holds for if and only if It is easy to check that which indicate that .

(ii) The necessary and sufficient conditions for the right-hand side inequality of (4.1) to hold are obvious.

(iii) As shown in Simpson and Spector [2], satisfies the Riccati equation Then where . Clearly, if is the unique solution of the equation on , then so is equation (4.2) on .

This completes the proof. □

Remark 4.2

Taking in Proposition 4.1 gives where the left-hand side inequality holds for and , the right-hand side one is inequality (1.4).

Setting in Proposition 4.1 yields where the left-hand side inequality is inequality (1.5).

In addition, putting with in Proposition 4.1, where is given in (1.7), we obtain a new Simpson–Spector-type inequality, which is stated as a corollary.

Corollary 4.3

Let . Then the double inequalities hold for . The lower and upper bounds are sharp.

Sharp bounds for in the form of

A bound in the form of for the ratio is known as Amos-type bound (see [6, 9, 10]). In this subsection, we will give another type of bounds in the form of for by Proposition 4.1. Clearly, .

As mentioned in Introduction, Baricz and Neuman [8, Theorem 2.2] (also see [6, Lemma 4.2.]) have shown that is strictly increasing from onto for . This implies that for , and then the double inequality of (4.1) is equivalent to for and with . Thus from Proposition 4.1 we derive the following statement.

Proposition 4.4

Let . (i) The double inequality (4.5) holds for and if and only if where is as in Proposition 4.1.

(ii) Furthermore, let Then we have Moreover, and are not comparable for .

(i) By Proposition 4.1, the necessary and sufficient condition such that the left-hand side inequality of (4.5) holds for and is clear.

While the right-hand side inequality of (4.5) holds for and if and only if where () are given in Proposition 4.1. Simplifying yields , , , , and which imply that .

(ii) To prove the second assertion of this proposition, we first note that the function is increasing on , and another function is decreasing on .

Now, since the function is increasing on and decreasing on , which implies .

If with , then , and then which is increasing in p on . This gives .

If with , then , and therefore, which is decreasing in p on . This leads to .

Finally, we show that is not comparable with for . In fact, we have that for , From this it is seen that if and if .

This completes the proof. □

Remark 4.5

Amos [10, Eq. (11)] offered a lower bound for and . Hornik and Grün [9, Theorem 6] showed that the Amos-type bound is the sharpest for and . Yang and Zheng [6, Theorem 4.6] extended the range of v from to . Proposition 4.4 presents another lower bound defined in (4.6) for for with and shows that defined by (4.8) is the maximum over all lower bounds . It should be emphasized that our sharpest lower bound extends the range of from to although and have the same expression.

Remark 4.6

Amos [10, Eq. (16)] gave an upper bound for and . Hornik and Grün [9, Theorem 3] proved that this Amos-type upper bound is the best for and , where the range of v has been extended from to in [6, Theorem 4.4] by Yang and Zheng. Since , our Proposition 4.4 demonstrates the same result in [6, Theorem 4.4] by a slightly different approach.

Remark 4.7

Proposition 4.4 also gives another upper bounds for by defined in (4.10) for and with , that is, Not only the above inequalities are valid, but we explain that is the minimum over all upper bounds , and and are not comparable in x on for . This indicates that for is indeed a new sharpest upper bound for . Consequently, Proposition 4.4 in fact offers a new type of bounds () for , which is clearly different from the Amos-type bound . Moreover, inequality (4.13) is sharp at in view of as .

As a direct consequence of Proposition 4.4, we have the following.

Corollary 4.8

If with , then the double inequality holds for all . Inequalities (4.14) are reversed if with .

In particular, taking , , and , −∞, we have

Remark 4.9

The right-hand side in inequality (4.15) for was proved in [10, Eq. (16)] by Amos, and for it follows from Neuman’s inequality (1.4). The right-hand side one in (4.16) for is also due to Amos [10, Eq. (11)], which for was proved by Yuan and Kalbfleisch [12, Eq. (A.5)], and Laforgia and Natalini [14, Theorem 1.1]. While the left-hand side inequality in (4.16) for was showed by Segura [15, Eq. (61)]. Inequalities (4.17) were proved by Yang and Zheng in [6, Remark 4.9]. Moreover, the rational bounds given in (4.18) appeared in [4, Appendix] for , so the right-hand side inequality of (4.18) can be viewed as a new one in the sense that the range of v is extended from to .

Now let us return to Proposition 4.1. First, if , then the right-hand side inequality of (4.1) holds for if and only if , which implies that the double inequality holds for . Second, according to the guess presented in Remark 3.1, may equal for certain with . If so, then the right-hand side inequality of (4.1) holds for if and only if , which implies that the double inequality (4.19) also holds for and certain with , where is the best possible constant. In fact, this claim is valid.

Proposition 4.10

Let . Then the double inequality (4.19) holds for and with the best constant .

(i) For , the desired result is evidently valid by Proposition 4.1.

(ii) For , where is defined by (1.7), it is easy to check that which imply . To prove the desired assertion, it suffices to prove that for and with . Indeed, we have and for , which yield In view of , the upper bound given in (4.20) is sharp, and by Proposition 4.1 the desired assertion follows. Thus we complete the proof. □

Remark 4.11

It is easy to check that the lower bound for given in (4.19) is weaker than , but the upper bound for is clearly a new Amos-type bound for with since it is not comparable with the sharpest upper bound for with , while another one is restricted in .

Some computable bounds for

From Proposition 4.4 we see that the minimum for with such that the inequality holds for . Since , where is the unique solution of equation (4.2) on , the number is usually not computable, and so is . Therefore, it is interesting and useful to find some upper bounds for by elementary functions.

In this subsection, we will find some upper bounds for in terms of elementary functions to obtain some computable upper bounds for by using relation (3.7), that is, and an analogous technique used in the proof of Subcase 2.4 of Theorem 1.1.

Proposition 4.12

Let with . Then the inequality holds for , where

It suffices to prove . We first prove that holds for all by dividing the proof into two cases.

Case 1. , namely . For this, we write as Then we have for .

Case 2. , namely . Similarly, we write as Then we get for .

Second, to prove that for all , we write in the form of Then, for , we have Finally, it is obtained that which completes the proof. □

Now by Proposition 4.12 we have the following.

Corollary 4.13

Let with .

For , the inequality holds for .

For , the inequality holds for . In particular, taking and letting with and with , the following inequalities hold for :

Remark 4.14

It is easy to check that the function defined in (4.22) is increasing on , which yields for with . This shows that the upper bound for is weaker than as the sharpest one given in Proposition 4.4. Inequality (4.26) seems to be a new one.

Remark 4.15

Clearly, for . In general, the upper bound for with given in (4.23) is not comparable with other two sharpest upper bounds for and for given in Proposition 4.4. For example, for , if and if , since

Similarly, for , if and if . In fact, some elementary computations give

It thus can be seen that the upper bound for with belongs to the new type of bounds () for .

Let us return to Proposition 4.1 again. We note that the number is also not computable in the case of If a better upper estimation holds for , then by Proposition 4.1 we can obtain some bounds similar to the double inequality (4.19), which also implies the new type of bounds. In fact, by the same technique as the proof of Proposition 4.12, we can prove the following proposition, but omit all the details of the proof.

Proposition 4.16

Let with . Then the double inequality holds for , where

From Proposition 4.10, we know the number for with . It remains to estimate for with . By a similar technique as the proof of Proposition 4.12, we have for with . Thus, by Proposition 4.1 we conclude the following proposition.

Proposition 4.17

Let with . Then the double inequality holds for , where is given in (4.28).

Remark 4.18

Similarly, the lower bounds given by (4.27) and (4.29) are trivial due to the fact that they are weaker than . However, the upper bounds are new ones which belong to the type of bounds ().

Conclusions

This paper is mainly devoted to proving the monotonicity of on for . As one of applications, from this we arrived at the Simpson–Spector-type inequalities for (4.1) and other new ones (Proposition 4.1 and Corollary 4.3), which immediately led to some known Simpson–Spector-type inequalities.

As more important applications, we reproved some known results and also found a new type of bounds for .

(i) Proposition 4.4 showed that the lower bound for is the sharpest, which for , and are known results (see [6, 9, 10]).

(ii) Proposition 4.4 also indicated that both the upper bounds for are the sharpest, where the former appeared in [6] and for and was proved in [9, 10], while the latter is a new comer and belongs to the type of bounds ().

(iii) We obtained in Proposition 4.10 a new Amos-type bound for , that is, holds for and with .

(iv) For with , the number given in Proposition 4.1 is in general not computable. But by replacing with defined by (4.21), we gave in Proposition 4.12 a class of elementary function upper bounds As mentioned in Remark 4.15, as an upper bound, is in general not comparable with other two sharpest upper bounds for and for , and belongs to the new type of bounds ().

(v) Using the same technique as Proposition 4.12, we established two new double inequalities for in the cases of and for , that are, (4.27) and (4.29). However, the lower bounds given in (4.27) and (4.29) for are trivial since they are weaker than . The upper bounds belong to the type of bounds ().

Additionally, as a consequence of our results, we deduced some new inequalities for , for example, (4.18), (4.26), and also reobtained some known important inequalities, such as the inequalities proved by Amos [10], Yuan and Kalbfleisch [12, (A.5)], Laforgia and Natalini [14, Theorem 1.1], Segura [15, (61)], [4, Appendix] and so on.