Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete ...

Feng Qi

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Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity

Feng Qi

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

Abstract

In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, 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 divided differences of polygamma functions, and related monotonicity results.

Cite this article:

  • Qi, Feng. "Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity." Turkish Journal of Analysis and Number Theory 2.5 (2014): 152-164.
  • Qi, F. (2014). Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity. Turkish Journal of Analysis and Number Theory, 2(5), 152-164.
  • Qi, Feng. "Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity." Turkish Journal of Analysis and Number Theory 2, no. 5 (2014): 152-164.

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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.

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