Review  Open  Published:
Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to ElezovićGiordanoPečarić’s theorem
Journal of Inequalities and Applicationsvolume 2013, Article number: 542 (2013)
Abstract
In the expository review and survey paper dealing with bounds for the ratio of two gamma functions, along one of the main lines of bounding the ratio of two gamma functions, the authors look back and analyze some known results, including Wendel’s asymptotic relation, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, Kershaw’s, and ElezovićGiordanoPečarić’s inequalities, LazarevićLupaş’s claim, and other monotonic and convex properties. On the other hand, the authors introduce some related advances on the topic of bounding the ratio of two gamma functions in recent years.
MSC: 33B15, 26A48, 26A51, 26D07, 26D15, 44A10.
1 Introduction
Recall [[1], Chapter XIII] and [[2], Chapter IV] that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and
for $x\in I$ and $n\ge 0$. The celebrated BernsteinWidder theorem [[2], p.160, Theorem 12a] states that a function f is completely monotonic on $[0,\mathrm{\infty})$ if and only if
where μ is a nonnegative measure on $[0,\mathrm{\infty})$ such that the integral (2) converges for all $x>0$. This tells us that a completely monotonic function f on $[0,\mathrm{\infty})$ is a Laplace transform of the measure μ.
It is well known that the classical Euler gamma function may be defined for $x>0$ by
The logarithmic derivative of $\mathrm{\Gamma}(x)$, denoted by $\psi (x)=\frac{{\mathrm{\Gamma}}^{\prime}(x)}{\mathrm{\Gamma}(x)}$, is called the psi or digamma function, and ${\psi}^{(k)}(x)$ for $k\in \mathbb{N}$ are called the polygamma functions. It is common knowledge that the special functions $\mathrm{\Gamma}(x)$, $\psi (x)$, and ${\psi}^{(k)}(x)$ for $k\in \mathbb{N}$ are fundamental and important and have many extensive applications in mathematical sciences.
The history of bounding the ratio of two gamma functions has been longer than sixty years since the paper [3] by Wendel was published in 1948. The motivations for bounding the ratio of two gamma functions are various, including establishment of asymptotic relation, refinements of Wallis’ formula, approximation to π, needs in statistics and other mathematical sciences.
In this review paper, along one of the main lines of bounding the ratio of two gamma functions, we would like to look back and analyze some known results, including Wendel’s asymptotic relation, Gurland’s approximation to π, Kazarinoff’s refinement of Wallis’ formula, Gautschi’s double inequality, Watson’s monotonicity, Chu’s refinement of Wallis’ formula, LazarevićLupaş’s claim on monotonic and convex properties, Kershaw’s first double inequality, ElezovićGiordanoPečarić’s theorem, alternative proofs of ElezovićGiordanoPečarić’s theorem and related consequences.
On the other hand, we would also like to describe some new advances in recent years on this topic, including the complete monotonicity of divided differences of the psi and polygamma functions, inequalities for sums and related results.
2 Wendel’s asymptotic relation
Our starting point is the paper published in 1948 by Wendel, which is the earliest related one we could search out to the best of our ability.
In order to establish the classical asymptotic relation
for real s and x, by using Hölder’s inequality for integrals, Wendel [3] proved elegantly the double inequality
for $0<s<1$ and $x>0$.
Wendel’s original proof Let
and apply Hölder’s inequality for integrals and the recurrence formula
for $x>0$ to obtain
Replacing s by $1s$ in (7), we get
from which we obtain
by substituting $x+s$ for x.
Combining (7) and (9), we get
Therefore, inequality (5) follows.
Letting x tend to infinity in (5) yields (4) for $0<s<1$. The extension to all real s is immediate on repeated application of (6). □
Remark 1 Inequality (5) can be rewritten for $0<s<1$ and $x>0$ as
or
Remark 2 Using recurrence formula (6) and double inequality (10) repeatedly yields
for $x>0$ and $0<s<1$, where m and n are positive integers. This implies that basing on recurrence formula (6) and double inequality (10), one can bound the ratio $\frac{\mathrm{\Gamma}(x+a)}{\mathrm{\Gamma}(x+b)}$ for any positive numbers x, a and b. Conversely, double inequality (12) reveals that one can also deduce corresponding bounds of the ratio $\frac{\mathrm{\Gamma}(x+1)}{\mathrm{\Gamma}(x+s)}$ for $x>0$ and $0<s<1$ from bounds of the ratio $\frac{\mathrm{\Gamma}(x+a)}{\mathrm{\Gamma}(x+b)}$ for positive numbers x, a, and b.
Remark 3 In [[4], p.257, 6.1.46], the following limit was listed: For real numbers a and b,
Limits (4) and (13) are equivalent to each other since
Hence, limit (13) is called Wendel’s asymptotic relation in the literature.
Remark 4 Double inequality (5) or (10) is more meaningful than limit (4) or (13), since the former implies the latter, but not conversely.
Remark 5 For more general results about the asymptotic expansion of a ratio of two gamma functions, please refer to [5, 6].
Remark 6 Due to unknown reasons, Wendel’s paper [3] and double inequality (5) or (10) were seemingly neglected by nearly all mathematicians for more than fifty years until it was mentioned in [7], to the best of our knowledge.
Remark 7 There has been a lot of literature about asymptotic expansions of a ratio of two gamma functions. Because this expository review and survey paper is devoted to inequalities and complete monotonicity, we will not further survey the topic of asymptotic expansions of a quotient of two gamma functions. For more information, please refer to [5, 8–14] and closely related references therein.
3 Gurland’s double inequality
By making use of a basic theorem in mathematical statistics concerning unbiased estimators with minimum variance, Gurland [15] presented the following inequality:
for $n\in \mathbb{N}$, and so taking respectively $n=2k$ and $n=2k+1$ for $k\in \mathbb{N}$ in (14) yields a closer approximation to π:
Remark 8 Taking respectively $n=2k$ and $n=2k1$ for $k\in \mathbb{N}$ in (14) leads to
This is better than double inequality (10) for $x=k$ and $s=\frac{1}{2}$.
Remark 9 Double inequality (15) may be rearranged as
It is easy to see that the upper bound in (17) is better than the corresponding one in (16). This phenomenon seemingly hints that sharper bounds for the ratio $\frac{\mathrm{\Gamma}(k+1)}{\mathrm{\Gamma}(k+1/2)}$ can be obtained only if letting $m\in \mathbb{N}$ in $n=2m1$ in (14). However, this is an illusion, since the lower bound of the following double inequality:
which is derived from taking respectively $n=2(k+m1)$ and $n=2(k+m1)1$ for $k\in \mathbb{N}$ in (14), is decreasing and the upper bound of it is increasing with respect to m. Then how can we explain the occurrence that the upper bound in (17) is stronger than the corresponding one in (16)?
Remark 10 The lefthand side inequality in (16) or (17) may be rearranged as
From this, it is easier to see that inequality (14) refines double inequality (10) for $x=k$ and $s=\frac{1}{2}$.
Remark 11 It is noted that inequality (14) was recovered in [16] and extended in [17] by different approaches respectively. See Section 4 and Section 7 below.
Remark 12 Just like the paper [3], Gurland’s paper [15] was ignored except that it was mentioned in [18, 19]. The famous monograph [20] recorded neither of the papers [3, 15].
4 Kazarinoff’s double inequality
Starting from one form of the celebrated formula of John Wallis,
which had been quoted for more than a century before 1950s by writers of textbooks, Kazarinoff proved in [17] that the sequence $\theta (n)$ defined by
satisfies $\frac{1}{4}<\theta (n)<\frac{1}{2}$ for $n\in \mathbb{N}$. This implies
Remark 13 It was said in [17] that it is unquestionable that inequalities similar to (22) can be improved indefinitely but at a sacrifice of simplicity, which is why inequality (20) had survived so long.
Remark 14 Kazarinoff’s proof of (22) is based upon the property
of the function
for $1<t<\mathrm{\infty}$. Inequality (23) was proved by making use of wellknown Legendre’s formula
for $x>0$ and estimating the integrals
Since (23) is equivalent to the statement that the reciprocal of $\varphi (t)$ has an everywhere negative second derivative, therefore, for any positive t, $\varphi (t)$ is less than the harmonic mean of $\varphi (t1)$ and $\varphi (t+1)$, which implies
As a subcase of this result, the righthand side inequality in (22) is established.
Remark 15 Using recurrence formula (6) in (27) gives
for $t>0$, which extends inequality (14). This shows that Kazarinoff’s paper [17] contains more general conclusions and that all results in [15] stated in paper [17] contain many general conclusions, and that all results in [15] are stated below.
Replacing t by 2t in (27) or (28) and rearranging yield
for $t>0$, which extends the lefthand side inequality in (16) and (17). Replacing t by $2t1$ in (27) or (28) produces
for $t>\frac{1}{2}$, which extends the righthand side inequality in (16). Replacing t by $2t+1$ in (27) or (28) and rearranging gives
for $t>\frac{1}{2}$, which extends the righthand side inequality in (17).
Remark 16 By the wellknown Wallis’ cosine formula [21], the sequence $\theta (n)$ defined by (21) may be rearranged as
for $n\in \mathbb{N}$. Then inequality (22) is equivalent to
Remark 17 Inequality (23) may be rewritten as
for $t>1$. Letting $u=\frac{t+1}{2}$ in the above inequality yields
for $u>0$. This inequality has been generalized in [22] to the complete monotonicity of a function involving divided differences of the digamma and trigamma functions as follows.
Theorem 1 [22]
For real numbers s, t, $\alpha =min\{s,t\}$, and λ, let
on $(\alpha ,\mathrm{\infty})$. Then the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ has the following complete monotonicity:

1.
For $0<ts<1$,

(a)
the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ is completely monotonic on $(\alpha ,\mathrm{\infty})$ if and only if $\lambda \le 1$,

(b)
so is the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ if and only if $\lambda \ge \frac{1}{ts}$;

(a)

2.
For $ts>1$,

(a)
the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ is completely monotonic on $(\alpha ,\mathrm{\infty})$ if and only if $\lambda \le \frac{1}{ts}$,

(b)
so is the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ if and only if $\lambda \ge 1$;

(a)

3.
For $s=t$, the function ${\mathrm{\Delta}}_{s,s;\lambda}(x)$ is completely monotonic on $(s,\mathrm{\infty})$ if and only if $\lambda \le 1$;

4.
For $ts=1$,

(a)
the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ is completely monotonic if and only if $\lambda <1$,

(b)
so is the function ${\mathrm{\Delta}}_{s,t;\lambda}(x)$ if and only if $\lambda >1$,

(c)
and ${\mathrm{\Delta}}_{s,t;1}(x)\equiv 0$.

(a)
Remark 18 Taking in Theorem 1 $\lambda =st>0$ produces that the function $\frac{\mathrm{\Gamma}(x+s)}{\mathrm{\Gamma}(x+t)}$ on $(t,\mathrm{\infty})$ is increasingly convex if $st>1$ and increasingly concave if $0<st<1$.
5 Watson’s monotonicity
In 1959, motivated by the result in [17] mentioned in Section 4, Watson [23] observed that
for $x>\frac{1}{2}$, which implies that the more general function
for $x>\frac{1}{2}$, whose special case is the sequence $\theta (n)$ for $n\in \mathbb{N}$ defined in (21) or (32), is decreasing and
This apparently implies the sharp inequalities
for $x>\frac{1}{2}$,
for $x\ge \frac{1}{4}$, and, by Wallis’ cosine formula [21],
In [23], an alternative proof of double inequality (40) was also provided as follows. Let
for $x>\frac{1}{2}$. By using the fairly obvious inequalities
and
we have, for $x>\frac{1}{4}$,
that is to say,
Remark 19 It is easy to see that inequality (41) extends and improves inequalities (10), (57), and (58) if $s=\frac{1}{2}$.
Remark 20 The lefthand side inequality in (42) is better than the corresponding one in (22) but worse than the corresponding one in (15) for $n\ge 2$.
Remark 21 Formula (37) implies the complete monotonicity of the function $\theta (x)$ defined by (38) on $(\frac{1}{2},\mathrm{\infty})$.
6 Gautschi’s double inequalities
The main aim of the paper [24] was to establish the double inequality
for $x\ge 0$ and $p>1$, where
or ${c}_{p}=1$.
By an easy transformation, inequality (47) was written in terms of the complementary gamma function
as
for $x\ge 0$ and $p>1$. In particular, letting $p\to \mathrm{\infty}$, the double inequality
for the exponential integral ${E}_{1}(x)=\mathrm{\Gamma}(0,x)$ for $x>0$ was derived from (50), in which the bounds exhibit the logarithmic singularity of ${E}_{1}(x)$ at $x=0$.
As a direct consequence of inequality (50) for $p=\frac{1}{s}$, $x=0$ and ${c}_{p}=1$, the following simple inequality for the gamma function was deduced:
The second main result of the paper [24] was a sharper and more general inequality
for $0\le s\le 1$ and $n\in \mathbb{N}$ than (52) by proving that the function
is monotonically decreasing for $0\le s<1$. Since $\psi (n)<lnn$, it was derived from inequality (53) that
which was also rewritten as
and so a simple proof of Euler’s product formula in the segment $0\le s\le 1$ was shown by letting $n\to \mathrm{\infty}$ in (56).
Remark 22 Double inequalities (53) and (55) can be further rearranged as
and
for $n\in \mathbb{N}$ and $0\le s\le 1$.
Remark 23 The upper bounds in (10) and (57) have the following relationship:
for $0\le s\le \frac{1}{2}$ and $n\in \mathbb{N}$, and inequality (59) reverses for $s>{e}^{1\gamma}1=0.52620\dots $, since the function
was proved in [[25], Theorem 2] to be strictly decreasing on $(1,\mathrm{\infty})$, with
This means that Wendel’s double inequality (10) and Gautschi’s first double inequality (57) are not included in each other but they all contain Gautschi’s second double inequality (58).
Remark 24 The righthand side inequality in (57) may be rearranged as
This suggests us the following double inequality:
for real numbers s, t and $x\in (min\{s,t\},\mathrm{\infty})$, where $\alpha (x)\sim x$ and $\beta (x)\sim x$ as $x\to \mathrm{\infty}$. For detailed information on the type of inequalities like (63), please refer to [26] and related references therein.
Remark 25 Inequality (58) can be rewritten as
for $n\in \mathbb{N}$ and $0\le s\le 1$.
Remark 26 In the reviews on the paper [24] by the Mathematical Reviews and the Zentralblatt MATH, there is not a word to comment on inequalities in (57) and (58). However, these two double inequalities later became a major source of a series of studies on bounding the ratio of two gamma functions.
7 Chu’s double inequality
In 1962, by discussing that
if and only if $(14c)n+13c\u2a8c0$, where
it was demonstrated in [[16], Theorem 1] that
As an application of (67), by using $\mathrm{\Gamma}(\frac{1}{2})=\sqrt{\pi}$ and (6), the following double inequality
for positive integers $n\ge 2$ was given in [[16], Theorem 2].
Remark 27 After letting $n=2k+1$, inequality (68) becomes
which is the same as (16). Taking $n=2k+2$ in (68) leads to inequalities (17) and (19).
Notice that the reasoning directions in the two papers [15, 16] are opposite:
To some extent, the results obtained by Gurland in [15] and by Chu in [16] are equivalent to each other, and they are all special cases of those obtained by Kazarinoff in [17].
Remark 28 By Wallis’ cosine formula [21], sequence (66) may be rewritten as
for $n\in \mathbb{N}$. Therefore, Chu discussed equivalently the necessary and sufficient conditions such that the sequence ${B}_{c}(n)$ for $n\in \mathbb{N}$ is monotonic.
Recently, necessary and sufficient conditions for the general function
on $(\rho ,\mathrm{\infty})$, where a, b, and c are real numbers and $\rho =min\{a,b,c\}$, to be logarithmically completely monotonic are presented in [27, 28]. A positive function f is said to be logarithmically completely monotonic on an interval $I\subseteq \mathbb{R}$ if it has derivatives of all orders on I and its logarithm lnf satisfies ${(1)}^{k}{[lnf(x)]}^{(k)}\ge 0$ for $k\in \mathbb{N}$ on I, see [29–31].
8 LazarevićLupaş’s claim
In 1974, among other things, the function
on $(0,\mathrm{\infty})$ for $\alpha \in (0,1)$ was claimed in [[32], Theorem 2] to be decreasing and convex, and so
Remark 29 The proof of [[32], Theorem 2] is wrong, see [[33], Remark 3.3] and [[34], p.240]. However, the statements in [[32], Theorem 2] are correct.
9 Kershaw’s first double inequality
In 1983, motivated by inequality (58) obtained in [24], among other things, Kershaw presented in [35] the following double inequality:
for $0<s<1$ and $x>0$. In the literature, it is called Kershaw’s first double inequality for the ratio of two gamma functions.
Kershaw’s proof for (75) Define the function ${g}_{\beta}$ by
for $x>0$ and $0<s<1$, where the parameter β is to be determined.
It is not difficult to show, with the aid of Wendel’s limit (13), that
To prove double inequality (75), define
from which it follows that
This leads to

1.
if $\beta =\frac{s}{2}$, then ${G}^{\prime}(x)<0$ for $x>0$;

2.
if $\beta =\frac{1}{2}+{(s+\frac{1}{4})}^{1/2}$, then ${G}^{\prime}(x)>0$ for $x>0$.
Consequently, if $\beta =\frac{s}{2}$, then G strictly decreases, and since $G(x)\to 1$ as $x\to \mathrm{\infty}$, it follows that $G(x)>1$ for $x>0$. However, from (77), this implies that ${g}_{\beta}(x)>{g}_{\beta}(x+1)$ for $x>0$, and so ${g}_{\beta}(x)>{g}_{\beta}(x+n)$. Take the limit as $n\to \mathrm{\infty}$ to give the result that ${g}_{\beta}(x)>1$, which can be rewritten as the lefthand side inequality in (75). The corresponding upper bound can be verified by a similar argument when $\beta =\frac{1}{2}+{(s+\frac{1}{4})}^{1/2}$, the only difference being that in this case ${g}_{\beta}$ strictly increases to unity. □
Remark 30 The spirit of Kershaw’s proof is similar to Chu’s in [[16], Theorem 1], as shown by (65). This idea or method was also utilized independently in [36–41] to construct, for various purposes, a number of inequalities of the type
for $s>0$ and real number $x\ge 0$.
Remark 31 It is easy to see that inequality (75) refines and extends inequalities (10) and (58).
Remark 32 Inequality (75) may be rearranged as
for $x>0$ and $0<s<1$.
10 ElezovićGiordanoPečarić’s theorem
Inequalities (11), (19), (64), and (80), sequence (32), and functions (38) and (73) strongly suggest us to consider the monotonic and convex properties of the general function
for $x\in (\alpha ,\mathrm{\infty})$, where s and t are two real numbers and $\alpha =min\{s,t\}$.
In 2000, Elezović, Giordano, and Pečarić gave in [[34], Theorem 1] a perfect solution to the monotonic and convex properties of the function ${z}_{s,t}(x)$ as follows.
Theorem 2 The function ${z}_{s,t}(x)$ is either convex and decreasing for $ts<1$ or concave and increasing for $ts>1$.
Remark 33 Direct computation yields
To prove the positivity of function (82), the following formula and inequality are used as basic tools in the proof of [[34], Theorem 1].

1.
For $x>1$,
$$\psi (x+1)=\gamma +\sum _{k=1}^{\mathrm{\infty}}(\frac{1}{k}\frac{1}{x+k}).$$(83) 
2.
If $a\le b<c\le d$, then
$$\frac{1}{ab}+\frac{1}{cd}>\frac{1}{ac}+\frac{1}{bd}.$$(84)
Remark 34 As consequences of Theorem 2, the following useful conclusions are derived.

1.
The function
$${e}^{\psi (x+t)}x$$(85)is decreasing and convex from $(t,\mathrm{\infty})$ onto $(t\frac{1}{2},t)$, where $t\in \mathbb{R}$.

2.
For all $x>0$,
$${e}^{\psi (x)}<\frac{1}{{\psi}^{\prime}(x)}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\psi (x)<ln{\psi}^{\prime}(x).$$(86) 
3.
For all $x>0$ and $t>0$,
$$ln(x+t\frac{1}{2})<\psi (x+t)<ln(x+{e}^{\psi (t)}).$$(87) 
4.
For $x>\alpha $, the inequality
$${\left[\frac{\mathrm{\Gamma}(x+t)}{\mathrm{\Gamma}(x+s)}\right]}^{1/(ts)}<\frac{ts}{\psi (x+t)\psi (x+s)}$$(88)
holds if $ts<1$ and reverses if $ts>1$.
Remark 35 In fact, function (85) is deceasing and convex on $(t,\mathrm{\infty})$ for all $t\in \mathbb{R}$. See [[25], Theorem 2].
Remark 36 It is clear that double inequality (87) can be deduced directly from the decreasingly monotonic property of (85). Furthermore, from the decreasingly monotonic and convex properties of (85) on $(t,\mathrm{\infty})$, inequality (86) and
on $(0,\mathrm{\infty})$ can be derived straightforwardly.
11 Recent advances
Finally, we would like to state some new results related to or originating from ElezovićGiordanoPečarić’s Theorem 2 above.
Alternative proofs of ElezovićGiordanoPečarić’s theorem
The key step of verifying Theorem 2 is to prove the positivity of the righthand side in (82), which involves divided differences of the digamma and trigamma functions. The biggest barrier or difficulty to prove the positivity of (82) is mainly how to deal with the squared term in (82).
Chen’s proof
In [42], the barrier mentioned above was overcome by virtue of the wellknown convolution theorem [43] for Laplace transforms, and so Theorem 2 for the special case $s+1>t>s\ge 0$ was proved. Perhaps this is the first try to provide an alternative of Theorem 2, although it was partially successful formally.
QiGuoChen’s proof
For real numbers α and β with $(\alpha ,\beta )\notin \{(0,1),(1,0)\}$ and $\alpha \ne \beta $, let
In [44, 45], by making use of the convolution theorem for the Laplace transform and the logarithmically convex properties of the function ${q}_{\alpha ,\beta}(x)$ on $(0,\mathrm{\infty})$, an alternative proof of Theorem 2 was supplied.
GuoQi’s first proof
In [46, 47], by considering monotonic properties of the function
and still employing the convolution theorem for the Laplace transform, Theorem 2 was completely verified again.
Remark 37 For more information on the function ${q}_{\alpha ,\beta}(t)$ and its applications, please refer to [26, 27, 46–51] and related references therein.
GuoQi’s second proof
In [52–55], the complete monotonic properties of the function on the righthand side of (82) were established as follows.
Theorem 3 Let s and t be two real numbers and $\alpha =min\{s,t\}$. Define
on $x\in (\alpha ,\mathrm{\infty})$. Then the functions ${\mathrm{\Delta}}_{s,t}(x)$ for $ts<1$ and ${\mathrm{\Delta}}_{s,t}(x)$ for $ts>1$ are completely monotonic on $x\in (\alpha ,\mathrm{\infty})$.
Since the complete monotonicity of the functions ${\mathrm{\Delta}}_{s,t}(x)$ and ${\mathrm{\Delta}}_{s,t}(x)$ mean the positivity and negativity of the function ${\mathrm{\Delta}}_{s,t}(x)$, an alternative proof of Theorem 2 was provided once again.
One of the key tools or ideas used in the proofs of Theorem 3 is the following simple but specially successful conclusion: If $f(x)$ is a function defined on an infinite interval $I\subseteq \mathbb{R}$ and it satisfies ${lim}_{x\to \mathrm{\infty}}f(x)=\delta $ and $f(x)f(x+\epsilon )>0$ for $x\in I$ and some fixed number $\epsilon >0$, then $f(x)>\delta $ on I.
It is clear that Theorem 3 is a generalization of inequality (89).
Complete monotonicity of divided differences
In order to prove Theorem 3, the following complete monotonic properties of a function related to a divided difference of the psi function were discovered in [54], the preprint of [55].
Theorem 4 Let s and t be two real numbers and $\alpha =min\{s,t\}$. Define
on $x\in (\alpha ,\mathrm{\infty})$. Then the functions ${\delta}_{s,t}(x)$ for $ts<1$ and ${\delta}_{s,t}(x)$ for $ts>1$ are completely monotonic on $x\in (\alpha ,\mathrm{\infty})$.
To the best of our knowledge, the complete monotonicity of functions involving divided differences of the psi and polygamma functions were investigated first in [52–57].
Inequalities for sums
As consequences of proving Theorem 4 along different approach from [55] and its preprint [54], the following algebraic inequalities for sums were procured in [56, 57] accidentally.
Theorem 5 Let k be a nonnegative integer and let $\theta >0$ be a constant.
If $a>0$ and $b>0$, then
holds for $ba>\theta $ and reverses for $ba<\theta $.
If $a<\theta $ and $b<\theta $, then the inequalities
and
hold for $ba>\theta $ and reverse for $ba<\theta $.
If $\theta <a<0$ and $\theta <b<0$, then inequality (95) holds and inequality (96) is valid for $a+b+\theta >0$ and is reversed for $a+b+\theta <0$.
If $a<\theta $ and $b>0$, then inequality (95) holds and inequality (96) is valid for $a+b+\theta >0$ and is reversed for $a+b+\theta <0$.
If $a>0$ and $b<\theta $, then inequality (95) is reversed and inequality (96) holds for $a+b+\theta <0$ and reverses for $a+b+\theta >0$.
If $b=a\theta $, then inequalities (94), (95) and (96) become equalities.
Moreover, the following equivalent relation between inequality (94) and Theorem 4 was found in [56, 57].
Theorem 6 Inequality (94) for positive numbers a and b is equivalent to Theorem 4.
Recent advances
Recently, some applications, extensions, and generalizations of Theorems 3 to 5, and related conclusions have been investigated in several recently or immediately published manuscripts such as [58–72]. For example, Theorem 1 stated in Remark 17 was obtained in [22].
The complete monotonicity of the qanalogue of the function ${\delta}_{0,0}$ defined by (93) was researched in [73, 74].
Remark 38 This article is a slightly revised version of the preprint [75] and a companion paper of the preprint [76] and the articles [26, 77] whose preprints are [78, 79], respectively.
References
 1.
Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht; 1993.
 2.
Widder DV: The Laplace Transform. Princeton University Press, Princeton; 1946.
 3.
Wendel JG: Note on the gamma function. Am. Math. Mon. 1948, 55(9):563–564. 10.2307/2304460
 4.
Abramowitz M, Stegun IA (Eds): National Bureau of Standards, Applied Mathematics Series 55 In Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington; 1970. 9th printing
 5.
Frame JS: An approximation to the quotient of gamma functions. Am. Math. Mon. 1949, 56(8):529–535. 10.2307/2305527
 6.
Tricomi FG, Erdélyi A: The asymptotic expansion of a ratio of gamma functions. Pac. J. Math. 1951, 1(1):133–142. Available online at http://projecteuclid.org/euclid.pjm/1102613160 10.2140/pjm.1951.1.133
 7.
Merkle M: Representations of error terms in Jensen’s and some related inequalities with applications. J. Math. Anal. Appl. 1999, 231: 76–90. 10.1006/jmaa.1998.6214
 8.
Burić T, Elezović N: Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions. J. Comput. Appl. Math. 2011, 235(11):3315–3331. 10.1016/j.cam.2011.01.045
 9.
Burić T, Elezović N: New asymptotic expansions of the quotient of gamma functions. Integral Transforms Spec. Funct. 2012, 23(5):355–368. 10.1080/10652469.2011.591393
 10.
Fields JL: A note on the asymptotic expansion of a ratio of gamma functions. Proc. Edinb. Math. Soc. 1966, 15(1):43–45. 10.1017/S0013091500013171
 11.
Fields JL: The uniform asymptotic expansion of a ratio of gamma functions. In Constructive Theory of Functions. Izdat. Bolgar. Akad. Nauk, Sofia; 1972:171–176. (Russian)
 12.
Laforgia A, Natalini P: On the asymptotic expansion of a ratio of gamma functions. J. Math. Anal. Appl. 2012, 389(2):833–837. 10.1016/j.jmaa.2011.12.025
 13.
Mortici C: The quotient of gamma functions by the psi function. Comput. Appl. Math. 2011, 30(3):627–638. 10.1590/S180703022011000300008
 14.
Olver FWJ: On an asymptotic expansion of a ratio of gamma functions. Proc. R. Ir. Acad. A 1995, 95(1):5–9.
 15.
Gurland J: On Wallis’ formula. Am. Math. Mon. 1956, 63: 643–645. 10.2307/2310591
 16.
Chu JT: A modified Wallis product and some applications. Am. Math. Mon. 1962, 69(5):402–404. 10.2307/2312135
 17.
Kazarinoff DK: On Wallis’ formula. Edinb. Math. Notes 1956, 1956(40):19–21.
 18.
Dutka J: On some gamma function inequalities. SIAM J. Math. Anal. 1985, 16: 180–185. 10.1137/0516013
 19.
Rao Uppuluri VR: On a stronger version of Wallis’ formula. Pac. J. Math. 1966, 19(1):183–187. Available online at http://projecteuclid.org/euclid.pjm/1102993964 10.2140/pjm.1966.19.183
 20.
Mitrinović DS: Analytic Inequalities. Springer, New York; 1970.
 21.
Weisstein, EW: Wallis Cosine Formula. From MathWorld  A Wolfram Web Resource. Available online at http://mathworld.wolfram.com/WallisFormula.html.
 22.
Qi, F, Guo, BN: Necessary and sufficient conditions for a function involving divided differences of the di and trigamma functions to be completely monotonic. arXiv:0903.3071. eprintatarXiv.org
 23.
Watson, GN: A note on gamma functions. Proc. Edinburgh Math. Soc. 11 (1958/1959), no. 2, Edinburgh Math Notes No. 42 (misprinted 41) (1959), 7–9
 24.
Gautschi W: Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 1959/60, 38: 77–81.
 25.
Qi, F, Guo, BN: Sharp inequalities for the psi function and harmonic numbers. arXiv:0902.2524. eprintatarXiv.org
 26.
Qi F: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010., 2010: Article ID 493058 10.1155/2010/493058
 27.
Qi F, Guo BN: Wendel’s and Gautschi’s inequalities: refinements, extensions, and a class of logarithmically completely monotonic functions. Appl. Math. Comput. 2008, 205(1):281–290. 10.1016/j.amc.2008.07.005
 28.
Qi F, Guo BN: WendelGautschiKershaw’s inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic. RGMIA Res. Rep. Coll. 2007., 10(1): Article ID 2. Available online at http://rgmia.org/v10n1.php
 29.
Atanassov RD, Tsoukrovski UV: Some properties of a class of logarithmically completely monotonic functions. C. R. Acad. Bulgare Sci. 1988, 41(2):21–23.
 30.
Berg C: Integral representation of some functions related to the gamma function. Mediterr. J. Math. 2004, 1(4):433–439. 10.1007/s0000900400226
 31.
Qi F, Guo BN: Complete monotonicities of functions involving the gamma and digamma functions. RGMIA Res. Rep. Coll. 2004, 7(1):63–72. Available online at http://rgmia.org/v7n1.php
 32.
Lazarević I, Lupaş A: Functional equations for Wallis and Gamma functions. Publ. Elektrotehn. Fak. Univ. Beograd. Ser. Electron. Telecommun. Automat 1974, 461–497: 245–251.
 33.
Alzer H: Sharp bounds for the ratio of q gamma functions. Math. Nachr. 2001, 222(1):5–14. 10.1002/15222616(200102)222:1<5::AIDMANA5>3.3.CO;2H
 34.
Elezović N, Giordano C, Pečarić J: The best bounds in Gautschi’s inequality. Math. Inequal. Appl. 2000, 3: 239–252. 10.7153/mia0326
 35.
Kershaw D: Some extensions of W. Gautschi’s inequalities for the gamma function. Math. Comput. 1983, 41: 607–611. 10.2307/2007697
 36.
Giordano C, Laforgia A: Inequalities and monotonicity properties for the gamma function. J. Comput. Appl. Math. 2001, 133: 387–396. 10.1016/S03770427(00)006592
 37.
Giordano C, Laforgia A, Pečarić J: Monotonicity properties for some functions involving the ratio of two gamma functions. In Funzioni Speciali e Applicazioni. Edited by: Bellacicco A, Laforgia A. Franco Angeli, Milano; 1998:35–42.
 38.
Giordano C, Laforgia A, Pečarić J: Unified treatment of GautschiKershaw type inequalities for the gamma function. J. Comput. Appl. Math. 1998, 99(1–2):167–175. Proc. VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997) 10.1016/S03770427(98)00154X
 39.
Laforgia A: Further inequalities for the gamma function. Math. Comput. 1984, 42(166):597–600. 10.2307/2007604
 40.
Lorch L: Inequalities for ultraspherical polynomials and the gamma function. J. Approx. Theory 1984, 40(2):115–120. 10.1016/00219045(84)900200
 41.
Palumbo B: A generalization of some inequalities for the gamma function. J. Comput. Appl. Math. 1998, 88(2):255–268. 10.1016/S03770427(97)001878
 42.
Chen CP: Monotonicity and convexity for the gamma function. J. Inequal. Pure Appl. Math. 2005., 6(4): Article ID 100. Available online at http://www.emis.de/journals/JIPAM/article574.html
 43.
Weisstein, EW: Laplace Transform. From MathWorld  A Wolfram Web Resource. Available online at http://mathworld.wolfram.com/LaplaceTransform.html.
 44.
Qi F, Guo BN, Chen CP: The best bounds in GautschiKershaw inequalities. Math. Inequal. Appl. 2006, 9(3):427–436. 10.7153/mia0941
 45.
Qi F, Guo BN, Chen CP: The best bounds in GautschiKershaw inequalities. RGMIA Res. Rep. Coll. 2005., 8(2): Article ID 17. Available online at http://rgmia.org/v8n2.php
 46.
Guo BN, Qi F: An alternative proof of ElezovićGiordanoPečarić’s theorem. Math. Inequal. Appl. 2011, 14(1):73–78. 10.7153/mia1406
 47.
Qi, F, Guo, BN: An alternative proof of ElezovićGiordanoPečarić’s theorem. arXiv:0903.1174. eprintatarXiv.org
 48.
Guo BN, Qi F: Properties and applications of a function involving exponential functions. Commun. Pure Appl. Anal. 2009, 8(4):1231–1249. 10.3934/cpaa.2009.8.1231
 49.
Qi F: Monotonicity and logarithmic convexity for a class of elementary functions involving the exponential function. RGMIA Res. Rep. Coll. 2006., 9(3): Article ID 3. Available online at http://rgmia.org/v9n3.php
 50.
Qi F: Threelogconvexity for a class of elementary functions involving exponential function. J. Math. Anal. Approx. Theory 2006, 1(2):100–103.
 51.
Qi F, Luo QM, Guo BN:The function $({b}^{x}{a}^{x})/x$: ratio’s properties. In Analytic Number Theory, Approximation Theory, and Special Functions. Edited by: Alladi K, Milovanović GV, Rassias MT. Springer, Berlin; 2014:673–682.
 52.
Guo BN, Qi F: A class of completely monotonic functions involving divided differences of the psi and trigamma functions and some applications. J. Korean Math. Soc. 2011, 48(3):655–667. 10.4134/JKMS.2011.48.3.655
 53.
Qi F: A completely monotonic function involving divided differences of psi and polygamma functions and an application. RGMIA Res. Rep. Coll. 2006., 9(4): Article ID 8. Available online at http://rgmia.org/v9n4.php
 54.
Qi F: The best bounds in Kershaw’s inequality and two completely monotonic functions. RGMIA Res. Rep. Coll. 2006., 9(4): Article ID 2. Available online at http://rgmia.org/v9n4.php
 55.
Qi F, Guo BN: Completely monotonic functions involving divided differences of the di and trigamma functions and some applications. Commun. Pure Appl. Anal. 2009, 8(6):1975–1989. 10.3934/cpaa.2009.8.1975
 56.
Qi F: A completely monotonic function involving divided difference of psi function and an equivalent inequality involving sum. RGMIA Res. Rep. Coll. 2006., 9(4): Article ID 5. Available online at http://rgmia.org/v9n4.php
 57.
Qi F: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums. ANZIAM J. 2007, 48(4):523–532. 10.1017/S1446181100003199
 58.
Guo BN, Qi F: A completely monotonic function involving the trigamma function and with degree one. Appl. Math. Comput. 2012, 218(19):9890–9897. 10.1016/j.amc.2012.03.075
 59.
Guo BN, Qi F: An extension of an inequality for ratios of gamma functions. J. Approx. Theory 2011, 163(9):1208–1216. 10.1016/j.jat.2011.04.003
 60.
Guo BN, Qi F: Monotonicity and logarithmic convexity relating to the volume of the unit ball. Optim. Lett. 2013, 7(6):1139–1153. 10.1007/s1159001204882
 61.
Guo BN, Qi F: Monotonicity of functions connected with the gamma function and the volume of the unit ball. Integral Transforms Spec. Funct. 2012, 23(9):701–708. 10.1080/10652469.2011.627511
 62.
Guo BN, Qi F: Refinements of lower bounds for polygamma functions. Proc. Am. Math. Soc. 2013, 141(3):1007–1015. 10.1090/S000299392012113875
 63.
Guo BN, Zhao JL, Qi F: A completely monotonic function involving the tri and tetragamma functions. Math. Slovaca 2013, 63(3):469–478. 10.2478/s1217501301092
 64.
Guo S, Xu JG, Qi F: Some exact constants for the approximation of the quantity in the Wallis’ formula. J. Inequal. Appl. 2013., 2013: Article ID 67 10.1186/1029242X201367
 65.
Qi F: Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities. Filomat 2013, 27(4):601–604. 10.2298/FIL1304601Q
 66.
Qi F, Berg C: Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function. Mediterr. J. Math. 2013, 10(4):1683–1694. 10.1007/s0000901302722
 67.
Qi F, Cerone P, Dragomir SS: Complete monotonicity of a function involving the divided difference of psi functions. Bull. Aust. Math. Soc. 2013, 88(2):309–319. 10.1017/S0004972712001025
 68.
Qi F, Guo BN: Necessary and sufficient conditions for functions involving the tri and tetragamma functions to be completely monotonic. Adv. Appl. Math. 2010, 44(1):71–83. 10.1016/j.aam.2009.03.003
 69.
Qi F, Luo QM, Guo BN: Complete monotonicity of a function involving the divided difference of digamma functions. Sci. China Math. 2013, 56(11):2315–2325. 10.1007/s1142501245620
 70.
Qi F, Wei CF, Guo BN: Complete monotonicity of a function involving the ratio of gamma functions and applications. Banach J. Math. Anal. 2012, 6(1):35–44.
 71.
Zhao JL, Guo BN, Qi F: A refinement of a double inequality for the gamma function. Publ. Math. (Debr.) 2012, 80(3–4):333–342. 10.5486/PMD.2012.5010
 72.
Zhao JL, Guo BN, Qi F: Complete monotonicity of two functions involving the tri and tetragamma functions. Period. Math. Hung. 2012, 65(1):147–155. 10.1007/s109980129562x
 73.
Qi, F: A completely monotonic function related to the qtrigamma function. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 76 (2014, in press)
 74.
Qi, F: Complete monotonicity of functions involving the qtrigamma and qtetragamma functions. arXiv:1301.0155
 75.
Qi, F: Bounds for the ratio of two gamma functionsFrom Wendel’s limit to ElezovićGiordanoPečarić’s theorem. arXiv:0902.2514. eprintatarXiv.org
 76.
Qi, F: Bounds for the ratio of two gamma functionsFrom Gautschi’s and Kershaw’s inequalities to completely monotonic functions. arXiv:0904.1049. eprintatarXiv.org
 77.
Qi F, Luo QM: Bounds for the ratio of two gamma functions  from Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 2012, 6(2):132–158.
 78.
Qi F: Bounds for the ratio of two gamma functions. RGMIA Res. Rep. Coll. 2008., 11(3): Article ID 1. Available online at http://rgmia.org/v11n3.php
 79.
Qi, F: Bounds for the ratio of two gamma functionsFrom Wendel’s and related inequalities to logarithmically completely monotonic functions. arXiv:0904.1048. eprintatarXiv.org
Acknowledgements
The original version of this article was reported on Thursday 24 July 2008 as a talk in the seminar held at the Research Group in Mathematical Inequalities and Applications (RGMIA), School of Computer Science and Mathematics, Victoria University, Australia, while the first author was visiting the RGMIA between March 2008 and February 2009 by the grant from the China Scholarship Council. The first author would like to express many thanks to Professors Pietro Cerone and Server S. Dragomir and other local colleagues for their invitation and hospitality throughout this period. This work was partially supported by the NNSF of China under Grant No. 11361038, by the NSF Project of Chongqing City under Grant No. CSTC2011JJA00024, by the Research Project of Science and Technology of Chongqing Education Commission under Grant No. KJ120625, and by the Fund of Chongqing Normal University, China under Grant No. 10XLR017 and 2011XLZ07, China. The authors appreciate three anonymous referees for their valuable comments on and careful corrections to the original version of this manuscript.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 bound
 inequality
 ratio of two gamma functions
 divided difference
 psi function
 polygamma function
 completely monotonic function
 Wallis’ formula
 inequality for sums