1. Introduction

For the sake of proceeding smoothly, we briefly introduce some necessary concepts and notation.

1.1. The Gamma and -gamma Functions

It is well-known that the classical Euler gamma function may be defined by

(1.1)

for . The logarithmic derivative of , denoted by , is called the psi or digamma function, and for are called the polygamma functions. It is common knowledge that special functions , and for are fundamental and important and have much extensive applications in mathematical sciences.

The -analogue of is defined [^[6], pp. 493-496] for by

(1.2)

(1.3)

The -gamma function has the following basic properties:

(1.4)

and

(1.5)

The -analogue of the psi or digamma function is defined by

for , where is a discrete measure with positive masses at the positive points for , more accurately,

(1.6)

See [^[33], p. 311] and its corrected version ^[34].

1.2. The Generalized Logarithmic Mean

The generalized logarithmic mean of order for positive numbers and with may be defined [^[13], p. 385] by

(1.7)

It is well-known that

(1.8)

(1.9)

and

(1.10)

are called respectively the geometric mean, the logarithmic mean, the identric or exponential mean, and the arithmetic mean. It is also known [^[13], pp. 386-387, Theorem 3] that the generalized logarithmic mean of order is increasing in for . Therefore, inequalities

(1.11)

are valid for and with . See also ^{[70, 71, 72, 115]}. Moreover, the generalized logarithmic mean is a special case of , that is, .

In passing, we remark that the complete monotonicity of the logarithmic mean was established in ^{[69, 84]}.

1.3. Logarithmically Completely Monotonic Functions

A function is said to be completely monotonic on an interval if has derivatives of all orders on and

(1.12)

for and .

Theorem 1.1. [^[118], p. 161] A necessary and sufficient condition that should be completely monotonic for is that

(1.13)

where is nondecreasing and the integral converges for .

Theorem 1.2. [^[11], p. 83] If is completely monotonic on , , and is completely monotonic on , then is completely monotonic on .

A positive function is said to be logarithmically completely monotonic on an interval if it has derivatives of all orders on and its logarithm satisfies

for on .

The notion “logarithmically completely monotonic function” was first put forward in ^[7] without an explicit definition. This terminology was explicitly recovered in ^[88] whose revised and expanded version was formally published as ^{[83, 90]}.

It has been proved once and again in ^{[10, 23, 66, 67, 83, 87, 88, 89, 103]} that a logarithmically completely monotonic function on an interval must also be completely monotonic on . C. Berg points out in ^[10] that these functions are the same as those studied by Horn ^[32] under the name infinitely divisible completely monotonic functions. For more information, please refer to ^{[10, 92, 93]} and related references therein.

1.4. Outline of this Paper

In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse Gautschi’s double inequality and Kershaw’s second double inequality, the complete monotonicity of several functions involving ratios of two gamma or -gamma functions by Alzer, Bustoz-Ismail, Elezović-Giordano-Pečarić and Ismail-Muldoon, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and the divided differences of polygamma functions, and related monotonicity results by Batir, Elezović-Pečarić, Qi and others.

2. Gautschi’s and Kershaw’s Double Inequalities

In this section, we begin with the papers ^{[22, 35]} to introduce a kind of inequalities for bounding the ratio of two gamma functions.

2.1. Gautschi’s Double Inequalities

The first result of the paper ^[22] was the double inequality

(2.1)

for and , where

(2.2)

or . By an easy transformation, the inequality (2.1) was written in terms of the complementary gamma function

(2.3)

as

(2.4)

for and . In particular, if letting , the double inequality

(2.5)

for the exponential integral for was derived from (2.4), in which the bounds exhibit the logarithmic singularity of at . As a direct consequence of the inequality (2.4) for and , the following simple inequality for the gamma function was deduced:

(2.6)

The second result of the paper ^[22] was a sharper and more general inequality

(2.7)

for and than (2.6). It was obtained by proving that the function

(2.8)

is monotonically decreasing for and that

Remark 2.1. For more information on refining the inequality (2.1), please refer to ^{[38, 96, 110]} and related references therein.

Remark 2.2. The left-hand side inequality in (2.7) can be rearranged as

(2.9)

or

(2.10)

for and Since the limit

(2.11)

can be verified by using Stirling’s formula in [1, p. 257, 6.1.38]: For there exists such that

(2.12)

it is natural to guess that the function

(2.13)

for is possibly increasing with respect to x on . This guess was verified and generalized in [^[52], Theorem 1], [^[53], Theorem 1], [^[85], Theorem 1], [^[86], Theorem 1] and others. See also Section 4.

Remark 2.3. For information on the study of the right-hand side inequality in (2.7), please refer to ^{[61, 62, 65, 105, 106]} and a great amount of related references therein.

2.2. Kershaw’s Second Double Inequality and Its Proof

In 1983, over twenty years later after the paper ^[22], among other things, D. Kershaw was motivated by the left-hand side inequality (2.7) in ^[22] and presented in ^[35] the following double inequality for and :

(2.14)

It is called in the literature Kershaw’s second double inequality.

Kershaw’s proof for (2.14). Define the function by

(2.15)

for and where the parameter is to be determined.

It is not difficult to show, with the aid of Stirling’s formula, that

(2.16)

Now let

(2.17)

Then

It is easy to show that

1. if then for ;

2. if , then for

Consequently if then strictly decreases, and since as it follows that for This implies that or and so Take the limit as to give the result that which can be rewritten as the left-hand side inequality in (2.14). The corresponding upper bound can be verified by a similar argument when the only difference being that in this case strictly increases to unity.

Remark 2.4. The idea contained in the above stated proof of (2.14) was also utilized by other mathematicians. For detailed information, please refer to related contents and references in ^{[61, 62]}.

Remark 2.5. The inequality (2.14) can be rearranged as

(2.18)

or

(2.19)

By Stirling’s formula (2.12), we can prove that

and

These clues make us to conjecture that the functions in the every end of inequalities (2.18) and (2.19) are perhaps monotonic with respect to x on

3. Several Complete Monotonicity Results

The complete monotonicity of the functions in the every end of inequalities (2.18) were first demonstrated in ^[12], and then several related functions were also proved in ^{[5, 19, 41]} to be (logarithmically) completely monotonic.

3.1. Bustoz-Ismail’s Complete Monotonicity Results

In 1986, motivated by the double inequality (2.14) and other related inequalities, J. Bustoz and M. E. H. Ismail revealed in [12, Theorem 7 and Theorem 8] that

1. the function [Trial mode]

(3.1)

for is completely monotonic on ; When the function (3.1) satisfies for

2. the function

(3.2)

for is strictly decreasing on

Remark 3.1. The proof of the complete monotonicity of the function (3.1) in [^[12], Theorem 7] relies on the inequality

(3.3)

for and the series representation

(3.4)

in [^[21], p. 15], and the above Theorem 1.2 applied to

Remark 3.2. The inequality (3.3) verified in [^[12], Lemma 3.1] can be rewritten as

(3.5)

for and , which is equivalent to

(3.6)

where stands for extended mean values and is defined for two positive numbers x and y and two real numbers r and s by

Actually, the inequality (3.6) is an immediate consequence of monotonicity of , see ^[39]. For more information, please refer to ^{[13, 17, 24, 29, 51, 57, 58, 72, 76, 79, 80, 91, 104, 107, 112, 113, 114, 119]} and related references therein.

Remark 3.3. The proof of the decreasing monotonicity of the function (3.2) just used the formula (3.4) and and the above Theorem 1.2 applied to

Remark 3.4. Indeed, J. Bustoz and M. E. H. Ismail had proved in [^[12], Theorem 7] that the function (3.1) is logarithmically completely monotonic on for . However, because the inequality (1.12) strictly holds for a completely monotonic function [Trial mode] on unless is constant (see [^[18], p. 98], [^[92], p. 82] and ^[117]), distinguishing between the cases and is not necessary.

3.2. Alzer’s and Related Complete Monotonicity Results

Stimulated by the complete monotonicity obtained in ^[12], including those mentioned above, H. Alzer obtained in [^[5], Theorem 1]] that the function

(3.7)

for and is completely monotonic on if and only if so is the reciprocal of (3.7) for and if and only if

As consequences of the monotonicity of the function (3.7), the following inequalities are deduced in [^[5], Corollary 2 and Corollary 3]:

1. The inequalities

(3.8)

for are valid for all and if and only if and

2. If

then

(3.9)

for

Remark 3.5. The inequality (3.9) follows from the formula

and the inequality (3.8) applied to and

Remark 3.6. The proof of the complete monotonicity of the function (3.7) in ^[5] is based on Theorem 1.2 applied to the formulas

(3.10)

and

(3.11)

for and discussing the positivity of the functions

(3.12)

for and Therefore, H. Alzer essentially gave in [^[5], Theorem 1] necessary and sufficient conditions for the function (3.7) to be logarithmically completely monotonic on

Remark 3.7. In [^[41], Theorem 3], a slight extension of [^[5], Theorem 1] was presented: The function

(3.13)

for and is logarithmically completely monotonic if and only if so is the reciprocal of (3.13) if and only if

The decreasing monotonicity of (3.13) and its reciprocal imply that the double inequality

(3.14)

for are valid for and if and only if and

It is obvious that the inequality (3.14) is a slight extension of the double inequality (3.8) obtained in [^[5], Corollary 2].

Remark 3.8. Specially we notice that [^[33], Theorem 3.4] has been corrected in [^[34], Theorem 3.4] as follows: Let and

where

Then the function is completely monotonic on for and the function is completely monotonic on for

As a consequence of [^[34], Theorem 3.4], the following result was deduced in [^[34], Corollary 3.5]: Let and

(3.15)

Then the function is completely monotonic on for the function is complete monotonic on for and neither is completely monotonic on for

Taking the limit in (3.15) yields [^[34], Corollary 3.6], a recovery, in a slightly extended form, of [^[5], Theorem 1] mentioned above.

The preprint ^[34] is a corrected version of the conference paper ^[33].

It is clear that [^[41], Theorem 3] can be derived by taking the limit

(3.16)

for

3.3. Ismail-Muldoon’s Complete Monotonicity Results

Inspired by inequalities (2.7) and (2.14), Ismail and Muldoon proved in [^[33], Theorem 3.2] the following conclusions: For and let

(3.17)

If then is completely monotonic on ; if then is completely monotonic on ; Neither or is completely monotonic for Consequently, the following inequality was deduced in [^[33], Theorem 3.3]: If the inequality

(3.18)

for holds for

Influenced by (3.18), H. Alzer posed in the final of the paper [^[4], p. 13] the following open problem: For real numbers and determine the best possible values and such that the inequalities

hold for all

Remark 3.9. Since the paper ^[33] was published in a conference proceedings, it is not easy to acquire it, so the completely monotonic properties of the function , obtained in [^[33], Theorem 3.2], were neglected in most circumstances.

3.4. Elezović-Giordano-Pečarić’s Inequality and Monotonicity Results

Inspired by the double inequality (2.14), the following problem was posed in [^[19], p. 247]: What are the best constants and such that the double inequality

(3.19)

holds for ?

An answer to the above problem was procured in [^[19], Theorem 4]: The double inequality

(3.20)

is valid for every and positive numbers and

Moreover, the function

(3.21)

for and was proved in [^[19], Theorem 5] to be completely monotonic on

Remark 3.10. It is clear that [^[19], Theorem 5] stated above extends or generalizes the complete monotonicity of the function (3.1).

Remark 3.11. By the way, the complete monotonicity in [^[19], Theorem 5] was iterated in [^[94], Proposition 4] and [^[95], Proposition 4] as follows: The function

(3.22)

is logarithmically completely monotonic with respect to on , where and are real numbers and

Remark 3.12. Along the same line as proving the inequality (3.20) in ^[19], the inequality (3.20) was generalized in [^[16], Theorem 2] as

(3.23)

for and where denotes the inverse function of

Remark 2.13. Since the inverse functions of the psi and polygamma functions are involved, it is much difficult to calculate the lower bounds in (3.20) and (3.23).

Remark 2.14. In ^[36], by the method used in ^[35], it was proved that the double inequality

(3.24)

holds for It s clear that the upper bound in (3.24) is a recovery of (3.20) and an immediate consequence of the complete monotonicity of the function (3.21).

4. Two Logarithmically Complete Monotonicity Results

Suggested by the double inequality (2.14), it is natural to put forward the following problem: What are the best constants and such that

(4.1)

is valid for ? where and are real numbers.

It is clear that the inequality (4.1) can also be rewritten as

(4.2)

which suggests some monotonic properties of the function

(4.3)

since the limit of the function (4.3) as is 1 by using (2.12).

This problem was considered in ^{[52, 53, 85, 86]} along two different approaches and the following results of different forms were established.

Theorem 4.1. [^[52], Theorem 1] and [^[53], Theorem 1] Let be real numbers and Define

for Furthermore, let be an implicit function defined by equation

(4.4)

on Then is decreasing and for and

1.  is logarithmically completely monotonic on if

2.  is logarithmically completely monotonic on if

Theorem 4.2. [^[85], Theorem 1] and [^[86], Theorem 1] For real numbers and with and a constant depending on and , define

(4.5)

1. The function is logarithmically completely monotonic on the interval if and only if ;

2. The function is logarithmically completely monotonic on the interval if and only if

Remark 4.1. In ^{[52, 53]}, it was deduced by standard argument that

for and . Therefore, the sufficient conditions in [^[52], Theorem 1] and [^[53], Theorem 1] are stated in terms of the implicit function [Trial mode] defined by (4.4).

Remark 4.2. In ^{[85, 86]}, the logarithmic derivative of was rearranged as

where

(4.6)

Since the function is increasing on with

and

the necessary and sufficient conditions in [^[85], Theorem 1] and [^[86], Theorem 1] may be derived immediately by considering Theorem 1.1.

However, the necessary conditions in [^[85], Theorem 1] and [^[86], Theorem 1] were proved by establishing the following inequalities involving the polygamma functions and their inverse functions in [^[85], Theorem 1] and [^[86], Theorem 1]:

1. If are two integers, then

(4.7)

where stands for the inverse function of for ;

2. The inequality

(4.8)

is valid for being positive odd number or zero and reversed for being positive even number;

3. The function

(4.9)

for is increasing and concave in and has a sharp upper bound

Note that if taking [Trial mode], [Trial mode], [Trial mode] and [Trial mode] in (4.7), (4.8) and (4.9), then [^[20], Lemma 1] and [^[20], Theorem 6] may be derived straightforwardly.

5. Recent Bounds and Monotonicity Results

In this section, we collect some recent bounds for the ratio of two gamma functions and gather several monotonicity results of functions involving the ratio of two gamma functions, divided differences of polygamma functions and mean values. Finally, we pose a conjecture.

5.1. Elezović-Pečarić’s Lower Bound

The inequality (4.8) for , that is, [^[20], Lemma 1], may be rewritten as

(5.1)

or, equivalently,

(5.2)

for positive numbers and .

Remark 5.1. From the left-hand side inequality in (1.11), it is easy to see that the inequality (5.2) refines the traditionally lower bound .

Remark 5.2. In [^[9], Theorem 2.4], the following incorrect double inequality was obtained:

(5.3)

where x and y are positive real numbers. Accurately speaking, the left-hand side inequality in (5.3) should be (5.2). See the first proof of [^[55], Theorem 1] or Section 5.5 below.

5.2. Allasia-Giordano-Pečarić’s Inequalities

In Section 4 of ^[3], as straightforward consequences of Hadamard type inequalities obtained in ^[2], the following double inequalities for bounding were listed: For , and , we have

and

where is an odd and positive integer,

(5.4)

and for are Bernoulli numbers defined by

If replacing by an even and positive integer, then the last four double inequalities are reversed.

5.3. Batir's Double Inequality for Polygamma Functions

It is clear that the double inequality (2.14) can be rearranged as

(5.5)

for and . The middle term in (5.5) can be regarded as a divided difference of the function on . Stimulated by this, N. Batir extended and generalized in [^[8], Theorem 2.7] the double inequality (5.5) as

(5.6)

where are positive numbers and .

5.4. Chen's Double Inequality in Terms of Polygamma Functions

In [^[15], Theorem 2], by virtue of the composite Simpson rule

in ^[31] and the formula

in ^[42], the following double inequalities and series representations were trivially shown: For and positive numbers and with ,

5.5. Recent Monotonicity Results by Qi and His Coauthors

Motivated by the left-hand side inequality in (5.3), although it is not correct, several refinements and generalizations about inequalities (5.2) and (5.6) were established by Qi and his coauthors in recent years.

5.5.1.

In [^[55], Theorem 1] and [^[56], Theorem 1], by virtue of the method used in [^[9], Theorem 2.4] and the inequality (4.8) for , the inequality (5.2) and the right-hand side inequality in (5.3) were recovered.

5.5.2.

In [^[55], Theorem 2] and [^[56], Theorem 2], the decreasing monotonicity of the function (3.2) and the right-hand side inequality in (3.20) were extended and generalized to the logarithmically complete monotonicity, and the inequality (5.2) was generalized to a decreasing monotonicity.

Theorem 5.1 ([^[55], Theorem 2] and [^[56], Theorem 2]).

For with , the function

(5.7)

is decreasing and

(5.8)

is logarithmically completely monotonic on , where

5.5.3.

In ^{[97, 98]}, the upper bounds in (2.14), (3.20), (5.3), (5.6) and related inequalities in ^{[52, 53, 85, 86]} were refined and extended as follows.

Theorem 5.2 (^{[97, 98]}). The inequalities

(5.9)

and

(5.10)

for and hold true.

Remark 5.3. The basic tools to prove (5.9) and (5.10) are an inequality in ^[14] and and a complete monotonicity in ^[101] respectively. They may be recited as follows:

1. If g is strictly monotonic, f is strictly increasing, and is convex (or concave, respectively) on an interval I, then

(5.11)

holds (or reverses, respectively) for . See also [^[13], p. 274, Lemma 2] and [^[20], p. 190, Theorem A].

2. The function

(5.12)

is completely monotonic on if and only if . See also ^{[99, 100]}.

Remark 5.4. By the so-called G-A convex approach, the inequality (5.9) was recovered in ^[120]: For ,

See also MR2413632, the review by MathSciNet of the paper ^[120]. Moreover, by the so-called geometrically convex method, the following double inequality was shown in [^[121], Theorem 1.2]: For positive numbers and ,

5.5.4.

In ^{[99, 100, 101]}, the function

(5.13)

was proved to be completely monotonic on if and only if . Utilizing the inequality (5.11) and the completely monotonic properties of the functions (5.12) and (5.13) yields the following double inequality.

Theorem 5.3 ([^[78], Theorem 1] and [^[102], Theorem 1]). For real numbers and with and an integer , the inequality

(5.14)

holds if and .

Remark 5.5. The double inequality (5.14) recovers, extends and refines inequalities (5.2), (5.6), (5.9) and (5.10).

Remark 5.6. A natural question is whether the above sufficient conditions and are also necessary for the inequality (5.14) to be valid.

5.5.5.

As generalizations of the inequalities (5.2), (5.6), the decreasing monotonicity of the function (5.7), and the left-hand side inequality in (5.14), the following monotonic properties were presented.

Theorem 5.4 ([^[78], Theorem 3] and [^[102], Theorem 3]). If is an integer, with , and, then the function

is increasing with respect to for either or and decreasing with respect to for, where .

Remark 5.7. It is not difficult to see that the ideal monotonic results of the function (5.15) should be stated as follows.

Conjecture 5.1. Let be an integer, with , and . Then the function (5.15) is increasing with respect to if and only if and decreasing with respect to if and only if .

Remark 5.8. Corresponding to Conjecture 5.1, the complete monotonicity of the function (5.15) and its negative may also be discussed.

Remark 5.9. This article is a slightly updated version of ^[63] and a companion paper of ^{[61, 105, 106]} and their preprints ^{[62, 64, 65]}.

Remark 5.10. Finally, we would like to recommend the articles [25,26,27,28,30,37,40,43-50,54,59,60,68,73,74,75,77,81,82,108,109,111,116] and closely related references therein to the readers for finding new developments and applications of the gamma function, polygamma functions, completely monotonic functions, logarithmically completely monotonic functions, concerned inequalities, asymptotic approximations, and so on.

Acknowledgements

The original version of this article was reported on 16 February 2009 as a talk in the seminar held at the RGMIA, School of Engineering and Science, Victoria University, Australia, while the author was visiting the RGMIA between March 2008 and February 2009 by the grant from the China Scholarship Council. The author expresses thanks to Professors Pietro Cerone and Server S. Dragomir and other local colleagues at Victoria University for their invitation and hospitality throughout this period.

References

[1]	M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington, 1970.
	In article
[2]	G. Allasia, C. Giordano, and J. Pečarić, Hadamard-type inequalities for (2r)-convex functions with applications, Atti Accad. Sci. Torino Cl. Sci. Fis Mat Natur. 133 (1999), 187-200.
	In article
[3]	G. Allasia, C. Giordano, and J. Pečarić, Inequalities for the gamma function relating to asymptotic expasions, Math. Inequal. Appl. 5 (2002), no. 3, 543-555.
	In article		CrossRef
[4]	H. Alzer, Sharp bounds for the ratio of q-gamma functions, Math. Nachr. 222 (2001), no. 1, 5-14.
	In article		CrossRef
[5]	H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337-346.
	In article		CrossRef
[6]	G. E. Andrews, R. A. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
	In article		CrossRef
[7]	R. D. Atanassov and U. V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), no. 2, 21-23.
	In article
[8]	N. Batir, On some properties of digamma and polygamma functions, J. Math. Anal. Appl. 328 (2007), no. 1, 452-465.
	In article		CrossRef
[9]	N. Batir, Some gamma function inequalities, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 5.
	In article
[10]	C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439.
	In article		CrossRef
[11]	S. Bochner, Harmonic Analysis and the Theory of Probability, California Monographs in Mathematical Sciences, University of California Press, Berkeley and Los Angeles, 1960.
	In article
[12]	J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), 659-667.
	In article		CrossRef
[13]	P. S. Bullen, Handbook of Means and Their Inequalities, Mathematics and its Applications, Volume 560, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003.
	In article		CrossRef
[14]	G. T. Cargo, Comparable means and generalized convexity, J. Math. Anal. Appl. 12 (1965), 387-392.
	In article		CrossRef
[15]	C.-P. Chen, On some inequalities for means and the second Gautschi-Kershaw’s inequality, RGMIA Res. Rep. Coll. 11 (2008), Suppl. Art. 6; Available online at http://rgmia.org/v11(E).php.
	In article
[16]	C.-P. Chen and A.-J. Li, Monotonicity results of integral mean and application to extension of the second Gautschi-Kershaw’s inequality, RGMIA Res. Rep. Coll. 10 (2007), no. 4, Art. 2.
	In article
[17]	C.-P. Chen and F. Qi, An alternative proof of monotonicity for the extended mean values, Aust. J. Math. Anal. Appl. 1 (2004), no. 2, Art. 11.
	In article
[18]	J. Dubourdieu, Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes, Compositio Math. 7 (1939-40), 96-111.
	In article
[19]	N. Elezović, C. Giordano, and J. Pečarić, The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000), 239-252.
	In article		CrossRef
[20]	N. Elezović and J. Pečarić, Differential and integral f-means and applications to digamma function, Math. Inequal. Appl. 3 (2000), no. 2, 189-196.
	In article		CrossRef
[21]	A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (Eds), Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.
	In article
[22]	W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959/60), 77-81.
	In article
[23]	B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21-30.
	In article
[24]	B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms 52 (2009), no. 1, 89-92.
	In article		CrossRef
[25]	B.-N. Guo and F. Qi, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Optim. Lett. 7 (2013), no. 6, 1139-1153.
	In article		CrossRef
[26]	B.-N. Guo and F. Qi, On the degree of the weighted geometric mean as a complete Bernstein function, Afr. Mat. 26 (2015), in press.
	In article		CrossRef
[27]	B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 1007-1015.
	In article		CrossRef
[28]	B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208.
	In article		CrossRef
[29]	B.-N. Guo and F. Qi, The function [Trial mode]: Logarithmic convexity and applications to extended mean values, Filomat 25 (2011), no. 4, 63-73.
	In article		CrossRef
[30]	B.-N. Guo, J.-L. Zhao, and F. Qi, A completely monotonic function involving the tri- and tetra-gamma functions, Math. Slovaca 63 (2013), no. 3, 469-478.
	In article		CrossRef
[31]	G. Hämmerlin and K.-H. Hoffmann, Numerical Mathematics, Translated from the German by Larry Schumaker. Undergraduate Texts in Mathematics. Readings in Mathematics. Springer, New York, 1991.
	In article		CrossRef
[32]	R. A. Horn, On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie und Verw. Geb 8 (1967), 219-230.
	In article		CrossRef
[33]	M. E. H. Ismail and M. E. Muldoon, Inequalities and monotonicity properties for gamma and q-gamma functions, in: R. V. M. Zahar (Ed.), Approximation and Computation: A Festschrift in Honour of Walter Gautschi, ISNM, 119, BirkhRauser, Basel, 1994, 309-323.
	In article		CrossRef
[34]	M. E. H. Ismail and M. E. Muldoon, Inequalities and monotonicity properties for gamma and q-gamma functions, available online at http://arxiv.org/abs/1301.1749.
	In article
[35]	D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983), 607-611.
	In article		CrossRef
[36]	D. Kershaw, Upper and lower bounds for a ratio involving the gamma function, Anal. Appl. (Singap. ) 3 (2005), no. 3, 293-295.
	In article		CrossRef
[37]	V. Krasniqi and F. Qi, Complete monotonicity of a function involving the [Trial mode]-psi function and alternative proofs, Glob. J. Math. Anal. 2 (2014), no. 3, 204-208.
	In article		CrossRef
[38]	A. Laforgia and P. Natalini, Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions, J. Inequal. Appl. 2006 (2006), Article ID 48727, 1-8.
	In article		CrossRef
[39]	E. B. Leach and M. C. Sholander, Extended mean values II, J. Math. Anal. Appl. 92 (1983), 207-223.
	In article		CrossRef
[40]	W.-H. Li, F. Qi, and B.-N. Guo, On proofs for monotonicity of a function involving the psi and exponential functions, Analysis (Munich) 33 (2013), no. 1, 45-50.
	In article		CrossRef
[41]	A.-J. Li, W.-Z. Zhao, and C.-P. Chen, Logarithmically complete monotonicity properties for the ratio of gamma function, Adv. Stud. Contemp. Math. (Kyungshang) 13 (2006), no. 2, 183-191.
	In article
[42]	E. Neuman and J. Sándor, On the Ky Fan inequality and related inequalities, II, Bull. Aust. Math. Soc. 72 (2005), no. 1, 87-107.
	In article		CrossRef
[43]	C. Mortici, A continued fraction approximation of the gamma function, J. Math. Anal. Appl. 402 (2013), no. 2, 405-410.
	In article		CrossRef
[44]	C. Mortici, A subtly analysis of Wilker inequality, Appl. Math. Comp. 231 (2014), 516-520.
	In article		CrossRef
[45]	C. Mortici, Estimating gamma function by digamma function, Math. Comput. Modelling 52 (2010), no. 5-6, 942-946.
	In article		CrossRef
[46]	C. Mortici, New approximation formulas for evaluating the ratio of gamma functions, Math. Comp. Modelling 52 (2010), no. 1-2, 425-433.
	In article		CrossRef
[47]	C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett. 23 (2010), no. 3, 97-100.
	In article		CrossRef
[48]	C. Mortici, New improvements of the Stirling formula, Appl. Math. Comput. 217 (2010), no. 2, 699-704.
	In article		CrossRef
[49]	C. Mortici, Ramanujan formula for the generalized Stirling approximation, Appl. Math. Comp. 217 (2010), no. 6, 2579-2585.
	In article		CrossRef
[50]	C. Mortici, The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl. 14 (2011), no. 3, 535-541.
	In article		CrossRef
[51]	J. Pečarić, F. Qi, V. Šimić and S.-L. Xu, Refinements and extensions of an inequality, III, J. Math. Anal. Appl. 227 (1998), no. 2, 439-448.
	In article		CrossRef
[52]	F. Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw’s inequality, J. Comput. Appl. Math. 224 (2009), no. 2, 538-543.
	In article		CrossRef
[53]	F. Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw’s inequality, RGMIA Res. Rep. Coll. 9 (2006), no. 4, Art. 11; Available online at http://rgmia.org/v9n4.php.
	In article
[54]	F. Qi, A completely monotonic function related to the q-trigamma function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 76 (2014), no. 1, 107-114.
	In article
[55]	F. Qi, A new lower bound in the second Kershaw’s double inequality, J. Comput. Appl. Math. 214 (2008), no. 2, 610-616.
	In article		CrossRef
[56]	F. Qi, A new lower bound in the second Kershaw’s double inequality, RGMIA Res. Rep. Coll. 10 (2007), no. 1, Art. 9; Available online at http://rgmia.org/v10n1.php.
	In article
[57]	F. Qi, A note on Schur-convexity of extended mean values, RGMIA Res. Rep. Coll. 4 (2001), no. 4, Art. 4, 529-533.
	In article
[58]	F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math. 35 (2005), no. 5, 1787-1793.
	In article		CrossRef
[59]	F. Qi, Absolute monotonicity of a function involving the exponential function, Glob. J. Math. Anal. 2 (2014), no. 3, 184-203.
	In article		CrossRef
[60]	F. Qi, An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind, J. Number Theory 144 (2014), 244-255.
	In article		CrossRef
[61]	F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages.
	In article		CrossRef
[62]	F. Qi, Bounds for the ratio of two gamma functions, RGMIA Res. Rep. Coll. 11 (2008), no. 3, Article 1.
	In article
[63]	F. Qi, Bounds for the ratio of two gamma functions—From Gautschi’s and Kershaw’s inequalities to completely monotonic functions, available online at http://arxiv.org/abs/0904.1049.
	In article
[64]	F. Qi, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, available online at http://arxiv.org/abs/0904.1048.
	In article
[65]	F. Qi, Bounds for the ratio of two gamma functions—From Wendel’s limit to Elezović-Giordano-Pečarić’s theorem, available online at http://arxiv.org/abs/0902.2514.
	In article
[66]	F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and q-gamma functions, Nonlinear Funct. Anal. Appl. 12 (2007), no. 4, 675-685.
	In article
[67]	F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and Q-gamma functions, RGMIA Res. Rep. Coll. 8 (2005), no. 3, Art. 5, 413-422.
	In article
[68]	F. Qi, Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions, Rev. R. Acad. Cienc. Exactas Fí s. Nat. Ser. A Math. RACSAM. 109 (2015).
	In article		CrossRef
[69]	F. Qi, Complete monotonicity of logarithmic mean, RGMIA Res. Rep. Coll. 10 (2007), no. 1, Art. 18; Available online at http://rgmia.org/v10n1.html.
	In article
[70]	F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math. 1 (2000), no. 1, Art. 4; Available online at http://www.emis.de/journals/JIPAM/article97.html.
	In article
[71]	F. Qi, Generalized abstracted mean values, RGMIA Res. Rep. Coll. 2 (1999), no. 5, Art. 4, 633-642.
	In article
[72]	F. Qi, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1978, 2723-2732.
	In article		CrossRef
[73]	F. Qi, Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function, J. Comput. Appl. Math. 268 (2014), 155-167.
	In article		CrossRef
[74]	F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory 133 (2013), no. 7, 2307-2319.
	In article		CrossRef
[75]	F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat 27 (2013), no. 4, 601-604.
	In article		CrossRef
[76]	F. Qi, Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1787-1796.
	In article		CrossRef
[77]	F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl. 18 (2015), in press; Available online at http://arxiv.org/abs/1302.6731.
	In article
[78]	F. Qi, Refinements, extensions and generalizations of the second Kershaw’s double inequality, RGMIA Res. Rep. Coll. 10 (2007), no. 2, Art. 8; Available online at http://rgmia.org/v10n2.php.
	In article
[79]	F. Qi, The extended mean values: Definition, properties, monotonicities, comparison, convexities, generalizations, and applications, Cubo Mat. Educ. 5 (2003), no. 3, 63-90.
	In article
[80]	F. Qi, The extended mean values: Definition, properties, monotonicities, comparison, convexities, generalizations, and applications, RGMIA Res. Rep. Coll. 5 (2002), no. 1, Art. 5, 57-80.
	In article
[81]	F. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10 (2013), no. 4, 1685-1696.
	In article		CrossRef
[82]	F. Qi, P. Cerone, and S. S. Dragomir, Complete monotonicity of a function involving the divided difference of psi functions, Bull. Aust. Math. Soc. 88 (2013), no. 2, 309-319.
	In article		CrossRef
[83]	F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607.
	In article		CrossRef
[84]	F. Qi and S.-X. Chen, Complete monotonicity of the logarithmic mean, Math. Inequal. Appl. 10 (2007), no. 4, 799-804.
	In article		CrossRef
[85]	F. Qi and B.-N. Guo, A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality, J. Comput. Appl. Math. 212 (2008), no. 2, 444-456.
	In article		CrossRef
[86]	F. Qi and B.-N. Guo, A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality, RGMIA Res. Rep. Coll. 10 (2007), no. 2, Art. 5; Available online at http://rgmia.org/v10n2.php.
	In article
[87]	F. Qi and B.-N. Guo, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, available online at http://arxiv.org/abs/0903.5038.
	In article
[88]	F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63-72.
	In article
[89]	F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, available online at http://arxiv.org/abs/0903.5123.
	In article
[90]	F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, 1283-1297.
	In article		CrossRef
[91]	F. Qi and B.-N. Guo, The function (bx-ax)/x: Logarithmic convexity and applications to extended mean values, available online at http://arxiv.org/abs/0903.1203.
	In article
[92]	F. Qi, B.-N. Guo, and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), 81-88.
	In article		CrossRef
[93]	F. Qi, B.-N. Guo, and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 5, 31-36.
	In article
[94]	F. Qi, B.-N. Guo, and C.-P. Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl. 9 (2006), no. 3, 427-436.
	In article		CrossRef
[95]	F. Qi, B.-N. Guo, and C.-P. Chen, The best bounds in Gautschi-Kershaw inequalities, RGMIA Res. Rep. Coll. 8 (2005), no. 2, Art. 17; Available online at http://rgmia.org/v8n2.php.
	In article
[96]	F. Qi and S. Guo, Inequalities for the incomplete gamma and related functions, Math. Inequal. Appl. 2 (1999), no. 1, 47-53.
	In article		CrossRef
[97]	F. Qi and S. Guo, New upper bounds in the second Kershaw’s double inequality and its generalizations, RGMIA Res. Rep. Coll. 10 (2007), no. 2, Art. 1; Available online at http://rgmia.org/v10n2.php.
	In article
[98]	F. Qi, S. Guo, and S.-X. Chen, A new upper bound in the second Kershaw’s double inequality and its generalizations, J. Comput. Appl. Math. 220 (2008), no. 1-2, 111-118.
	In article		CrossRef
[99]	F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149-2160.
	In article		CrossRef
[100]	F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, available online at http://arxiv.org/abs/0905.2732.
	In article
[101]	F. Qi, S. Guo, and B.-N. Guo, Note on a class of completely monotonic functions involving the polygamma functions, RGMIA Res. Rep. Coll. 10 (2007), no. 1, Art. 5; Available online at http://rgmia.org/v10n1.php.
	In article
[102]	F. Qi, X.-A. Li, and S.-X. Chen, Refinements, extensions and generalizations of the second Kershaw’s double inequality, Math. Inequal. Appl. 11 (2008), no. 3, 457-465
	In article		CrossRef
[103]	F. Qi, W. Li, and B.-N. Guo, Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 15; Available online at http://rgmia.org/v9n3.php.
	In article
[104]	F. Qi and Q.-M. Luo, A simple proof of monotonicity for extended mean values, J. Math. Anal. Appl. 224 (1998), no. 2, 356-359.
	In article		CrossRef
[105]	F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132-158.
	In article
[106]	F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem, J. Inequal. Appl. 2013, 2013:542, 20 pages.
	In article		CrossRef
[107]	F. Qi and Q.-M. Luo, Refinements and extensions of an inequality, Mathematics and Informatics Quarterly 9 (1999), no. 1, 23-25.
	In article
[108]	F. Qi, Q.-M. Luo, and B.-N. Guo, Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math. 56 (2013), no. 11, 2315-2325.
	In article		CrossRef
[109]	F. Qi, Q.-M. Luo, and B.-N. Guo, The function (bx-ax)/x: Ratio’s properties, In: Analytic Number Theory, Approximation Theory, and Special Functions, G. V. Milovanović and M. Th. Rassias (Eds), Springer, 2014, pp. 485-494.
	In article		CrossRef
[110]	F. Qi and J.-Q. Mei, Some inequalities of the incomplete gamma and related functions, Z. Anal. Anwendungen 18 (1999), no. 3, 793-799.
	In article		CrossRef
[111]	F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal. 2 (2014), no. 3, 91sh-97.
	In article		CrossRef
[112]	F. Qi and S.-L. Xu, Refinements and extensions of an inequality, II, J. Math. Anal. Appl. 211 (1997), 616-620.
	In article		CrossRef
[113]	F. Qi and S.-L. Xu, The function (bx-ax)/x: Inequalities and properties, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3355-3359.
	In article		CrossRef
[114]	F. Qi, S.-L. Xu, and L. Debnath, A new proof of monotonicity for extended mean values, Intern. J. Math. Math. Sci. 22 (1999), no. 2, 417-421.
	In article		CrossRef
[115]	F. Qi and S.-Q. Zhang, Note on monotonicity of generalized weighted mean values, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1989, 3259-3260.
	In article		CrossRef
[116]	F. Qi and X.-J. Zhang, Complete monotonicity of a difference between the exponential and trigamma functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 21 (2014), no. 2, 141-145.
	In article		CrossRef
[117]	H. van Haeringen, Completely monotonic and related functions, J. Math. Anal. Appl. 204 (1996), no. 2, 389-408.
	In article		CrossRef
[118]	D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
	In article
[119]	S.-L. Zhang, C.-P. Chen, and F. Qi, Another proof of monotonicity for the extended mean values, Tamkang J. Math. 37 (2006), no. 3, 207-209.
	In article		CrossRef
[120]	X.-M. Zhang and Y.-M. Chu, An inequality involving the Gamma function and the psi function, Int. J. Mod. Math. 3 (2008), no. 1, 67-73.
	In article
[121]	X.-M. Zhang, T.-Q. Xu, and L.-B. Situ, Geometric convexity of a function involving gamma function and applications to inequality theory, J. Inequal. Pure Appl. Math. 8 (2007), no. 1, Art. 17.
	In article