Abstract and Applied Analysis

Volume 2014 (2014), Article ID 702718, 9 pages

http://dx.doi.org/10.1155/2014/702718

Research Article

A Double Inequality for the Trigamma Function and Its Applications

Zhen-Hang Yang,¹ Yu-Ming Chu,¹ and Xiao-Jing Tao²

¹School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

²College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 19 March 2014; Accepted 26 May 2014; Published 16 July 2014

Academic Editor: Wing-Sum Cheung

Copyright © 2014 Zhen-Hang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove that and are the best possible parameters in the interval such that the double inequality holds for . As applications, some new approximation algorithms for the circumference ratio and Catalan constant are given. Here, is the trigamma function.

1. Introduction

For real and positive values of , the classical Euler’s gamma function and its logarithmic derivative , the so-called psi function, are defined as

For extension of these functions to complex variables and for basic properties, see [1]. The derivatives are known as polygamma functions (see [2]). In particular, is called trigamma function.

Recently, the bounds for the trigamma function using exponential functions have attracted the attention of many researchers. For example, Elezović et al. [3] proved that the inequality

holds for all . In [4, Theorem 2.7], Batir proved that and are the best possible constants such that the double inequality

holds for all , where is Euler’s constant. Batir [4] also showed that

for all . In [5, (1.11)], Guo and Qi established that

if and , . They [6, Lemma 2] found a very simple upper bound for trigamma function in terms of exponential function as follows:

for all . The inequality (6) was generalized in [7, Theorem 3.1], [8, Theorem 1.1], and [9, Theorem 1.1] to a complete monotonicity which reads that the difference is completely monotonic on . Many other new results involving the psi and trigamma functions can be found in the literature [10, 11].

Suppose that and and are the real functions defined on . is said to be the best possible constant in such that the inequality holds for all if on , or and are not comparable on , and for any satisfies on the interval .

The main purpose of this paper is to find the best possible constants such that the double inequality

or equivalently

holds for all , where

Our main result is the following Theorem 1.

Theorem 1. and are the best possible constants in the interval such that the double inequality (7) or (8) holds for all .

From Theorem 1, we clearly see the following.

Corollary 2. The double inequality

holds for all .

2. Lemmas

Lemma 3. Let the function be defined on by (9). Then the function is strictly decreasing with respect to on and strictly increasing on .

Proof. It follows from (9) that

If , then ; that is, is strictly increasing with respect to . Therefore,

which implies that .

If , then ; that is, is strictly decreasing with respect to , which leads to the conclusion that

Lemma 4. Let the function be defined on by (9) and

Then the equation

has two roots

such that for and for .

Proof. Differentiation yields

which reveals that the function is strictly increasing on and strictly decreasing on . Therefore, Lemma 4 follows from the piecewise monotonicity of the function and the numerical computations results:

Lemma 5. Let , , and and let be defined as in Lemma 4. Then the following statements are true:

(i)if the inequality holds for all , then ;(ii)if the inequality holds for all , then .

Proof. It follows from the series formulas that

and we get

(i)If inequality holds for all , then, from

and Lemma 4, we clearly see that .

 (ii)If inequality holds for all , then

and Lemma 4 lead to the conclusion that

Lemma 6. Let the function be defined on by (9). Then and are not comparable for all if .

Proof. For and , let

Then simple computation leads to

From (20) and (21), we have

Differentiation yields

which shows that is strictly decreasing on . Therefore,

if . On the other hand, we clearly see that for .

Lemma 7. Let and with and let be the polynomial of degree defined by

where and for with . Then, there exists such that and for and for .

Proof. Differentiating gives

Note that

for .

From , we clearly see that is strictly increasing on . Then and lead to the conclusion that there exists such that and for and for . Therefore, is strictly decreasing on and strictly increasing on .

It follows from the piecewise monotonicity of and that for , and there exists such that and for and for . Therefore, is strictly decreasing on and strictly increasing on .

After repeating the same steps as above times, we deduce that there exists such that and for and for .

Lemma 8. Let the function be defined on by (9). Then there exists such that and are not comparable for all if , and for all if .

Proof. For and , let

Then simple computation leads to

(i) We prove that there exists such that and are not comparable for all if . For this end, it suffices to prove that there exists such that and if .

Indeed, it follows from

that the function is strictly increasing on . Numerical computations show that

Therefore, there exists such that

and for and for .

On the other hand, it follows from together with (20) and (21) that

(ii)  We prove that for all if . From Lemma 3, we know that is strictly decreasing on , so it suffices to prove that for all .

Let

Then

Differentiation yields

where

Note that

where the second equation in (46) follows from (38).

Differentiating leads to

where

We assert that there exists a unique such that for and for , which leads to the conclusion that is strictly decreasing on and strictly increasing on . To this end, it is enough to verify that the coefficients of satisfy the conditions of Lemma 7. In fact, it follows from that we have

From the piecewise monotonicity of together with (46) and (47), we clearly see that

that is, for . Then (43) and (45) lead to the conclusion that for , which implies that is strictly increasing on and for .

Therefore, follows easily from (40) and .

Lemma 9 (see [12, pp. 258–260]). Let and . Then

From the proof of [4, Theorem 2.6], we get the following.

Lemma 10. The inequality

holds for .

The following lemma can be derived immediately from the proof of [4, Theorem 2.1].

Lemma 11 (see [4, Theorem 2.1]). Let be the function defined on by

Then for .

The well-known Hermite-Hadamard inequality for convex function can be stated as follows.

Lemma 12 (see [13]). Let be an interval, with , and let be a convex function. Then

3. Proofs of Theorem 1

Proof of Theorem 1. We divide the proof into four parts.

(I) We prove the first inequality in (7); that is,

where is defined by (14).

It follows from Lemma 10 that

where

We clearly see that it is enough to prove that for .

Differentiating gives

where denotes the logarithmic mean of positive numbers and , and

Differentiating leads to

for , which means that is strictly increasing on and . It in turn implies that is strictly decreasing on . Therefore, for .

(II) We prove that is the best possible constant such that for all .

From Lemma 5, we know that if for all , where . It follows from Lemma 6 that and are not comparable for all if ; that is to say, is not a better lower bound of than even if there exists such that .

For any , (22) leads to

It follows from (19), (20), and (21) that we get

(III) We prove the second inequality (7); that is,

for , where is defined by (14).

Lemma 11 implies that the function is strictly convex on . Then, making use of Lemma 12, we get

for . That is,

Therefore, inequality (64) follows easily from (52) and the last inequality above.

(IV) We prove that is the best possible constant such that for all .

From Lemma 5, we know that if for all , where .

It follows from Lemma 8 that there exists such that and are not comparable for all if and for all if . Lemma 3 leads to the conclusion that for all if .

If there exists such that for all , then, from (19), (20), (21), and (22), we get

From the above proof and Lemma 3 we get the following.

Corollary 13. Let the function be defined on by (9) and let be the root of (38) on . Then the inequalities

or equivalently

hold for all .

4. Remarks

Remark 14. It follows from (67) and the facts that

that we clearly see that the upper bound in Theorem 1 for the trigamma function is better than the upper bounds given in (2), (3), (4), (5), and (6) if is large enough.

Lemma 15. One has

Proof. Differentiation leads to

If , then, from , we get , which leads to

If , then the first inequality in (71) still holds by making a change of variable . If , since , we see that for and for . Hence,

Remark 16. Using inequality (71), one has

for , which shows that the upper bound in (8) is better than the upper bound in (6).

Remark 17. The conclusion that the difference is completely monotonic on given in [6, Lemma 2] implies that

or

It is easy to check that the lower bound and for given in (10) are not comparable due to

Remark 18. Guo et al. [14] proved that

for if . In particular, if , one has

We clearly see that the upper bound given in (69) is better than that in (80) for the trigamma function .

Finally, we give remarks on two mathematical constants and (Catalan constant).

Remark 19. It is well known that

Let , and then . From Theorem 1, we clearly see that the double inequality

holds for all .

Let

Then inequalities (82) can be rewritten as

or

It follows from (63) and (67) that

Therefore, (85) provides a new approximation algorithm for . Numerical simulations results carried out with mathematical software show that the given algorithm is more accurate than (see Table 1).

More estimate methods for can be found in [15–19].

Table 1

1
0.6921

2
0.40298

5
0.1782

10

50

100

200

500

Remark 20. The Catalan constant

is a mysterious constant in mathematics and physics. From and (see [17, 20]) together with Theorem 1, we get

for .

Let

Then inequalities (88) can be rewritten as

It follows from (63) and (67) that we have

Therefore, (90) provides a new approximation algorithm for the Catalan constant . Numerical simulations results carried out with mathematical software show that the given algorithm is more accurate than (see Table 2).

Table 2

1

2

5

10

50

100

200

500

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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