Open Access

A Class of Logarithmically Completely Monotonic Functions Associated with a Gamma Function

Journal of Inequalities and Applications20102010:392431

https://doi.org/10.1155/2010/392431

Received: 15 November 2010

Accepted: 27 December 2010

Published: 30 December 2010

Abstract

We show that the function is strictly logarithmically completely monotonic on if and only if and is strictly logarithmically completely monotonic on if and only if .

1. Introduction

For real and positive values of the Euler gamma function and its logarithmic derivative , the so-called digamma function, are defined as
(1.1)

For extension of these functions to complex variables and for basic properties, see [1]. These functions play central roles in the theory of special functions and have lots of extensive applications in many branches, for example, statistics, physics, engineering, and other mathematical sciences. Over the past half century monotonicity properties of these functions have attracted the attention of many authors (see [222]).

Recall that a real-valued function is said to be completely monotonic on if has derivatives of all orders on and
(1.2)

for all and . Moreover, is said to be strictly completely monotonic if inequality (1.2) is strict.

Recall also that a positive real-valued function is said to be logarithmically completely monotonic on if has derivatives of all orders on and its logarithm satisfies
(1.3)

for all and . Moreover, is said to be strictly logarithmically completely monotonic if inequality (1.3) is strict.

Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. In particular, many remarkable results for the complete monotonicity or logarithmically complete monotonicity involving the gamma, psi and polygamma functions can be found in the literature [18, 19, 2342].

The Kershaw's inequality in [21] states that the double inequality
(1.4)
holds for and . In [43], Laforgia extends the both sides of inequality in (1.4) as follows:
(1.5)

for or and , and inequality (1.5) is reversed for and .

Let us define
(1.6)

for with and . In order to establish the best bounds in Kershaw's inequality (1.4), the following monotonicity and convexity properties of are established in [13, 44, 45]: the function is either convex and decreasing for or concave and increasing for .

This work is motivated by an paper of Guo [46], who proved that the function
(1.7)
is strictly logarithmically concave and strictly increasing from onto . It is natural to ask for an extension of this result: is logarithmically complete monotonic? We will give the positive answer. Actually, we investigate a more general problem. The goal of this article is to discuss the logarithmically complete monotonicity properties of the functions
(1.8)

on and for fixed .

Recently Chen et al. [38, Theorem 1] proved the following result: let and be real numbers, define for ,
(1.9)

Then, the function is strictly logarithmically completely monotonic on if and only if . So is the function if and only if .

Our main results are summarized as follows.

Theorem 1.1.

Let , , and is defined as (1.8), then

(1) is strictly logarithmically completely monotonic on if and only if ;

(2) is strictly logarithmically completely monotonic on if and only if .

As applications of Theorem 1.1, one has the following corollaries.

Corollary 1.2.

For and , one has the double inequalities for the ratio of the gamma functions
(1.10)
In particular, one has
(1.11)
for and , and
(1.12)

for and .

Corollary 1.3.

For and , one has the following double inequality
(1.13)

where .

2. Lemmas

In order to prove our Theorem 1.1, we need serval lemmas which we collect in this section. In our second lemma we present the area of to determine positive (or negative) for a function, which plays a crucial role in the proof of our result Theorem 1.1 given in Section 3.

Let be a function defined on as
(2.1)
We will discuss the properties for this function and refer to view Figure 1 more clearly.
Figure 1

The shading area is denoted by . Otherwise, . The red curve is the graph of with an asymptotic line .

The function can be interpreted as a quadric equation with respect to , that is
(2.2)
where , , and its discriminant function
(2.3)

Obviously, if , then . It follows from that .

If , then . We can solve two roots of the equation , which are
(2.4)

It follows from the properties of the quadratic equation that for and for or .

Differentiating with respect to , one has
(2.5)

By (2.5) we know that the minimal value of can be attained at , that is . Moreover, is strictly decreasing on and strictly increasing on .

Obviously, is strictly increasing on . Note that
(2.6)

as . In other words, and has the asymptotic line .

Lemma 2.1.

The psi or digamma function, the logarithmic derivative of the gamma function, and the polygamma functions can be expressed as
(2.7)
(2.8)

for and , where is Euler's constant.

Lemma 2.2.

Let and . Then the following statements are true:

(1)if , then for ;

(2)if , then for ;

(3)if and , then there exist such that for and for .

Proof.

Let and . Then simple computations lead to
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
  1. (1)

    If , then we divide the proof into two cases.

     

Case 1.

If , then implies that for and for . Thus is strictly increasing on and strictly decreasing on . From (2.10) and we clearly see that there exists such that for and for , which implies that is strictly increasing on and strictly decreasing on . It follows from (2.9) that
(2.16)

for .

Case 2.

If , then we know since . It follows from (2.13) and (2.15) that
(2.17)

Therefore, there exists such that for and for follows from (2.17), which implies that is strictly decreasing on and strictly increasing on . It follows from (2.12) and that there exists such that for and for . By the same argument, it follows from (2.10) and that there exists such that for and for .

Therefore, for follows from (2.9).

  1. (2)

    If , then from Figure 1 we know that could be positive or negative. We divide the proof into two cases.

     

Case 1.

If , then from (2.13) and (2.15) we clearly know that for , which implies that is strictly increasing on . Then the properties of and lead to
(2.18)
It follows from (2.12) and (2.18) that there exists such that for and for since as . Hence is strictly decreasing on and strictly increasing on . From (2.10) and we know that there exists such that for and for . Therefore, it follows from (2.9) that
(2.19)

for .

Case 2.

If , then from (2.13) and (2.15) we know that there exists such that for and for . Thus is strictly decreasing on and strictly increasing on . It follows from (2.12) and that there exists such that for and for . By the same argument as Case 1, for follows from (2.9) and (2.10).

  1. (3)

    If and , then from (2.12) we clearly know that . Thus there exists such that for . It follows from (2.10) that , . Since as , we know that there exists such that for , which implies that is strictly increasing on and . Therefore, for and for .

     

We state a simple lemma as the results of [12, 47].

Lemma 2.3.

Inequality
(2.20)

holds for .

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Taking the logarithm of (1.8) and differentiating, then we have
(3.1)
For , it follows from (2.8) that
(3.2)
where
(3.3)
  1. (1)
    If , then it follows from (3.1) and (2.20) that
    (3.4)
     
From (3.2) and (3.3) together with Lemma 2.2(1) we clearly see that
(3.5)

holds for . Therefore, is strictly logarithmically completely monotonic on that follows from (3.4) and (3.5).

Conversely, if , then we can divide the set into two subsets: and . Therefore, it follows from Lemma 2.2(2) and (3) that is not strictly logarithmically completely monotonic on for .
  1. (2)
    If , then from (3.1) and (2.20) we get
    (3.6)
     
since
(3.7)
For , it follows from (3.2) that
(3.8)

where is defined as (3.3).

Therefore, is strictly logarithmically completely monotonic on that follows from (3.6), (3.8), and Lemma 2.2(2).

Conversely, if and , then we can divide the set into two subsets: and . Therefore, it follows from Lemma 2.2(1) and (3) that is not strictly logarithmically completely monotonic on for .

Remark 1.

Although the upper and lower bounds of Kershaw's inequalities given in (1.11) and (1.12) are not better than those of inequalities in (1.4) and (1.5), the difference between them is close to zero as is large enough. For example,
(3.9)

Furthermore, the advantage of our inequalities is to give the upper and lower bounds of Kershaw's inequality for and while Laforgia established only one side of Kershaw's inequality.

Declarations

Acknowledgment

This work was supported by the National Science Foundation of China under Grant no. 11071069 and the Natural Science Foundation of Zhejiang Province under Grant no. Y7080106.

Authors’ Affiliations

(1)
Department of Mathematics, Huzhou Teachers College

References

  1. Whittaker ET, Watson GN: A Course of Modern Analysis. Cambridge University Press, Cambridge, UK; 1996:vi+608.View ArticleMATHGoogle Scholar
  2. Chu YM, Zhang XM, Zhang Z: The geometric convexity of a function involving gamma function with applications. Korean Mathematical Society 2010, 25(3):373–383. 10.4134/CKMS.2010.25.3.373MathSciNetView ArticleMATHGoogle Scholar
  3. Zhang XM, Chu YM: A double inequality for gamma function. Journal of Inequalities and Applications 2009, 2009:-7.Google Scholar
  4. Zhao T-H, Chu Y-M, Jiang Y-P: Monotonic and logarithmically convex properties of a function involving gamma functions. Journal of Inequalities and Applications 2009, 2009:-13.Google Scholar
  5. Zhang XM, Chu YM: An inequality involving the gamma function and the psi function. International Journal of Modern Mathematics 2008, 3(1):67–73.MathSciNetMATHGoogle Scholar
  6. Chu YM, Zhang XM, Tang X: An elementary inequality for psi function. Bulletin of the Institute of Mathematics. Academia Sinica 2008, 3(3):373–380.MathSciNetMATHGoogle Scholar
  7. Song YQ, Chu YM, Wu L: An elementary double inequality for gamma function. International Journal of Pure and Applied Mathematics 2007, 38(4):549–554.MathSciNetMATHGoogle Scholar
  8. Chen Ch-P: Monotonicity and convexity for the gamma function. Journal of Inequalities in Pure and Applied Mathematics 2005, 6(4):6, article no. 100.MathSciNetMATHGoogle Scholar
  9. Guo B-N, Qi F: Two new proofs of the complete monotonicity of a function involving the PSI function. Bulletin of the Korean Mathematical Society 2010, 47(1):103–111. 10.4134/BKMS.2010.47.1.103MathSciNetView ArticleMATHGoogle Scholar
  10. Chen Ch-P, Qi F, Srivastava HM: Some properties of functions related to the gamma and psi functions. Integral Transforms and Special Functions 2010, 21(1–2):153–164.MathSciNetView ArticleMATHGoogle Scholar
  11. Qi F: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums. The ANZIAM Journal 2007, 48(4):523–532. 10.1017/S1446181100003199MathSciNetView ArticleMATHGoogle Scholar
  12. Qi F, Guo B-N: Monotonicity and convexity of ratio between gamma functions to different powers. Journal of the Indonesian Mathematical Society 2005, 11(1):39–49.MathSciNetMATHGoogle Scholar
  13. Chen Ch-P, Qi F: Inequalities relating to the gamma function. The Australian Journal of Mathematical Analysis and Applications 2004, 1(1, article no. 3):-7.Google Scholar
  14. Guo B-N, Qi F: Inequalities and monotonicity for the ratio of gamma functions. Taiwanese Journal of Mathematics 2003, 7(2):239–247.MathSciNetMATHGoogle Scholar
  15. Qi F: Monotonicity results and inequalities for the gamma and incomplete gamma functions. Mathematical Inequalities & Applications 2002, 5(1):61–67.MathSciNetView ArticleMATHGoogle Scholar
  16. Qi F, Mei J-Q: Some inequalities of the incomplete gamma and related functions. Zeitschrift für Analysis und Ihre Anwendungen 1999, 18(3):793–799.MathSciNetView ArticleMATHGoogle Scholar
  17. Qi F, Guo S-L: Inequalities for the incomplete gamma and related functions. Mathematical Inequalities & Applications 1999, 2(1):47–53.MathSciNetView ArticleMATHGoogle Scholar
  18. Alzer H: Some gamma function inequalities. Mathematics of Computation 1993, 60(201):337–346. 10.1090/S0025-5718-1993-1149288-7MathSciNetView ArticleMATHGoogle Scholar
  19. Alzer H: On some inequalities for the gamma and psi functions. Mathematics of Computation 1997, 66(217):373–389. 10.1090/S0025-5718-97-00807-7MathSciNetView ArticleMATHGoogle Scholar
  20. Anderson GD, Qiu S-L: A monotoneity property of the gamma function. Proceedings of the American Mathematical Society 1997, 125(11):3355–3362. 10.1090/S0002-9939-97-04152-XMathSciNetView ArticleMATHGoogle Scholar
  21. Kershaw D: Some extensions of W. Gautschi's inequalities for the gamma function. Mathematics of Computation 1983, 41(164):607–611.MathSciNetMATHGoogle Scholar
  22. Merkle M: Logarithmic convexity and inequalities for the gamma function. Journal of Mathematical Analysis and Applications 1996, 203(2):369–380. 10.1006/jmaa.1996.0385MathSciNetView ArticleMATHGoogle Scholar
  23. Alzer H, Berg Ch: Some classes of completely monotonic functions. II. The Ramanujan Journal 2006, 11(2):225–248. 10.1007/s11139-006-6510-5MathSciNetView ArticleMATHGoogle Scholar
  24. Alzer H: Sharp inequalities for the digamma and polygamma functions. Forum Mathematicum 2004, 16(2):181–221. 10.1515/form.2004.009MathSciNetView ArticleMATHGoogle Scholar
  25. Alzer H, Batir N: Monotonicity properties of the gamma function. Applied Mathematics Letters 2007, 20(7):778–781. 10.1016/j.aml.2006.08.026MathSciNetView ArticleMATHGoogle Scholar
  26. Clark WE, Ismail MEH: Inequalities involving gamma and psi functions. Analysis and Applications 2003, 1(1):129–140. 10.1142/S0219530503000041MathSciNetView ArticleMATHGoogle Scholar
  27. Elbert Á, Laforgia A: On some properties of the gamma function. Proceedings of the American Mathematical Society 2000, 128(9):2667–2673. 10.1090/S0002-9939-00-05520-9MathSciNetView ArticleMATHGoogle Scholar
  28. Bustoz J, Ismail MEH: On gamma function inequalities. Mathematics of Computation 1986, 47(176):659–667. 10.1090/S0025-5718-1986-0856710-6MathSciNetView ArticleMATHGoogle Scholar
  29. Ismail MEH, Lorch L, Muldoon ME: Completely monotonic functions associated with the gamma function and its -analogues. Journal of Mathematical Analysis and Applications 1986, 116(1):1–9. 10.1016/0022-247X(86)90042-9MathSciNetView ArticleMATHGoogle Scholar
  30. Babenko VF, Skorokhodov DS: On Kolmogorov-type inequalities for functions defined on a semi-axis. Ukrainian Mathematical Journal 2007, 59(10):1299–1312.MathSciNetView ArticleMATHGoogle Scholar
  31. Muldoon ME: Some monotonicity properties and characterizations of the gamma function. Aequationes Mathematicae 1978, 18(1–2):54–63. 10.1007/BF01844067MathSciNetView ArticleMATHGoogle Scholar
  32. Qi F, Yang Q, Li W: Two logarithmically completely monotonic functions connected with gamma function. Integral Transforms and Special Functions 2006, 17(7):539–542. 10.1080/10652460500422379MathSciNetView ArticleMATHGoogle Scholar
  33. Qi F, Niu D-W, Cao J: Logarithmically completely monotonic functions involving gamma and polygamma functions. Journal of Mathematical Analysis and Approximation Theory 2006, 1(1):66–74.MathSciNetMATHGoogle Scholar
  34. Qi F, Chen S-X, Cheung W-S: Logarithmically completely monotonic functions concerning gamma and digamma functions. Integral Transforms and Special Functions 2007, 18(5–6):435–443.MathSciNetView ArticleMATHGoogle Scholar
  35. Qi F: A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality. Journal of Computational and Applied Mathematics 2007, 206(2):1007–1014. 10.1016/j.cam.2006.09.005MathSciNetView ArticleMATHGoogle Scholar
  36. Chen Ch-P, Qi F: Logarithmically complete monotonicity properties for the gamma functions. The Australian Journal of Mathematical Analysis and Applications 2005, 2(2, article no. 8):-9.Google Scholar
  37. Chen Ch-P, Qi F: Logarithmically completely monotonic functions relating to the gamma function. Journal of Mathematical Analysis and Applications 2006, 321(1):405–411. 10.1016/j.jmaa.2005.08.056MathSciNetView ArticleMATHGoogle Scholar
  38. Chen Ch-P, Wang G, Zhu H: Two classes of logarithmically completely monotonic functions associated with the gamma function. Computational Intelligence Foundations and Applications 2010, 4: 168–173.View ArticleGoogle Scholar
  39. Chen Ch-P: Complete monotonicity properties for a ratio of gamma functions. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 2005, 16: 26–28.View ArticleMathSciNetMATHGoogle Scholar
  40. Merkle M: On log-convexity of a ratio of gamma functions. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 1997, 8: 114–119.MathSciNetMATHGoogle Scholar
  41. Li A-J, Chen Ch-P: Some completely monotonic functions involving the gamma and polygamma functions. Journal of the Korean Mathematical Society 2008, 45(1):273–287. 10.4134/JKMS.2008.45.1.273MathSciNetView ArticleMATHGoogle Scholar
  42. Qi F, Niu D-W, Cao J, Chen S-X: Four logarithmically completely monotonic functions involving gamma function. Journal of the Korean Mathematical Society 2008, 45(2):559–573. 10.4134/JKMS.2008.45.2.559MathSciNetView ArticleMATHGoogle Scholar
  43. Laforgia A: Further inequalities for the gamma function. Mathematics of Computation 1984, 42(166):597–600. 10.1090/S0025-5718-1984-0736455-1MathSciNetView ArticleMATHGoogle Scholar
  44. Elezović N, Giordano C, Pečarić J: The best bounds in Gautschi's inequality. Mathematical Inequalities & Applications 2000, 3(2):239–252.MathSciNetView ArticleMATHGoogle Scholar
  45. Qi F, Guo B-N, Chen Ch-P: The best bounds in Gautschi-Kershaw inequalities. Mathematical Inequalities & Applications 2006, 9(3):427–436.MathSciNetView ArticleMATHGoogle Scholar
  46. Guo S: Monotonicity and concavity properties of some functions involving the gamma function with applications. Journal of Inequalities in Pure and Applied Mathematics 2006, 7(2, article no. 45):-7.Google Scholar
  47. Qi F, Guo B-N: A new proof of complete monotonicity of a function involving psi function. RGMIA Research Report Collection 2008., 11(3, article no. 12):Google Scholar

Copyright

© T.-H. Zhao and Y.-M. Chu. 2010

This article is published under license to BioMed Central Ltd. 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.