Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem

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 Elezović-Giordano-Pečarić’s inequalities, Lazarević-Lupaş’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.


Introduction
where μ is a nonnegative measure on [, ∞) such that the integral () converges for all x > . This tells us that a completely monotonic function f on [, ∞) is a Laplace transform of the measure μ.
It is well known that the classical Euler gamma function may be defined for x >  by The logarithmic derivative of (x), denoted by ψ(x) = (x) (x) , is called the psi or digamma function, and ψ (k) (x) for k ∈ N are called the polygamma functions. It is common knowl-©2013 Qi and Luo; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.journalofinequalitiesandapplications.com/content/2013/1/542 edge that the special functions (x), ψ(x), and ψ (k) (x) for k ∈ 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 [] by Wendel was published in . 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, Lazarević-Lupaş's claim on monotonic and convex properties, Kershaw's first double inequality, Elezović-Giordano-Pečarić's theorem, alternative proofs of Elezović-Giordano-Pečarić'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.

Wendel's asymptotic relation
Our starting point is the paper published in  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 [] proved elegantly the double inequality for  < s <  and x > .

Wendel's original proof Let
and apply Hölder's inequality for integrals and the recurrence formula Replacing s by s in (), we get from which we obtain by substituting x + s for x. Combining () and (), we get Therefore, inequality () follows.
Letting x tend to infinity in () yields () for  < s < . The extension to all real s is immediate on repeated application of ().
Remark  Inequality () can be rewritten for  < s <  and x >  as Remark  Using recurrence formula () and double inequality () repeatedly yields for x >  and  < s < , where m and n are positive integers. This implies that basing on recurrence formula () and double inequality (), one can bound the ratio (x+a) (x+b) for any positive numbers x, a and b. Conversely, double inequality () reveals that one can also deduce corresponding bounds of the ratio (x+) (x+s) for x >  and  < s <  from bounds of the ratio (x+a) (x+b) for positive numbers x, a, and b.
Remark  In [, p., ..], the following limit was listed: For real numbers a and b, Limits () and () are equivalent to each other since Hence, limit () is called Wendel's asymptotic relation in the literature. http://www.journalofinequalitiesandapplications.com/content/2013/1/542 Remark  Double inequality () or () is more meaningful than limit () or (), since the former implies the latter, but not conversely.

Gurland's double inequality
By making use of a basic theorem in mathematical statistics concerning unbiased estimators with minimum variance, Gurland [] presented the following inequality: for n ∈ N, and so taking respectively n = k and n = k +  for k ∈ N in () yields a closer approximation to π : () Remark  Taking respectively n = k and n = k - for k ∈ N in () leads to This is better than double inequality () for x = k and s =   .
Remark  Double inequality () may be rearranged as It is easy to see that the upper bound in () is better than the corresponding one in (). This phenomenon seemingly hints that sharper bounds for the ratio (k+) (k+/) can be obtained only if letting m ∈ N in n = m - in (). However, this is an illusion, since the http://www.journalofinequalitiesandapplications.com/content/2013/1/542 lower bound of the following double inequality: which is derived from taking respectively n = (k + m -) and n = (k + m -) - for k ∈ N in (), 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 () is stronger than the corresponding one in ()?
Remark  The left-hand side inequality in () or () may be rearranged as From this, it is easier to see that inequality () refines double inequality () for x = k and s =   .

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 s by writers of textbooks, Kazarinoff proved in [] that the sequence θ (n) defined by Remark  It was said in [] that it is unquestionable that inequalities similar to () can be improved indefinitely but at a sacrifice of simplicity, which is why inequality () had survived so long. http://www.journalofinequalitiesandapplications.com/content/2013/1/542 Remark  Kazarinoff 's proof of () is based upon the property for - < t < ∞. Inequality () was proved by making use of well-known Legendre's formula for x >  and estimating the integrals Since () is equivalent to the statement that the reciprocal of φ(t) has an everywhere negative second derivative, therefore, for any positive t, φ(t) is less than the harmonic mean of φ(t -) and φ(t + ), which implies As a subcase of this result, the right-hand side inequality in () is established.
Remark  Using recurrence formula () in () gives for t > , which extends the left-hand side inequality in () and (). Replacing t by t - in () or () produces for t >   , which extends the right-hand side inequality in (). Replacing t by t +  in () or () and rearranging gives for t > -  , which extends the right-hand side inequality in ().
Remark  By the well-known Wallis' cosine formula [], the sequence θ (n) defined by () may be rearranged as for n ∈ N. Then inequality () is equivalent to Remark  Inequality () may be rewritten as for t > -. Letting u = t+  in the above inequality yields for u > . This inequality has been generalized in [] to the complete monotonicity of a function involving divided differences of the digamma and trigamma functions as follows.

Theorem  []
For real numbers s, t, α = min{s, t}, and λ, let . Then the function s,t;λ (x) has the following complete monotonicity: Remark  Taking in Theorem  λ = st >  produces that the function (x+s) (x+t) on (-t, ∞) is increasingly convex if st >  and increasingly concave if  < st < .

Watson's monotonicity
In , motivated by the result in [] mentioned in Section , Watson [] observed that for x > -  , which implies that the more general function for x > -  , whose special case is the sequence θ (n) for n ∈ N defined in () or (), is decreasing and This apparently implies the sharp inequalities for x ≥ -  , and, by Wallis' cosine formula [], In [], an alternative proof of double inequality () was also provided as follows. Let for x >   . By using the fairly obvious inequalities and that is to say, Remark  It is easy to see that inequality () extends and improves inequalities (), (), and () if s =   .
Remark  The left-hand side inequality in () is better than the corresponding one in () but worse than the corresponding one in () for n ≥ .

Gautschi's double inequalities
The main aim of the paper [] was to establish the double inequality for x ≥  and p > , where or c p = . By an easy transformation, inequality () was written in terms of the complementary gamma function for x ≥  and p > . In particular, letting p → ∞, the double inequality for the exponential integral E  (x) = (, x) for x >  was derived from (), in which the bounds exhibit the logarithmic singularity of E  (x) at x = . http://www.journalofinequalitiesandapplications.com/content/2013/1/542 As a direct consequence of inequality () for p =  s , x =  and c p = , the following simple inequality for the gamma function was deduced: The second main result of the paper [] was a sharper and more general inequality for  ≤ s ≤  and n ∈ N than () by proving that the function is monotonically decreasing for  ≤ s < . Since ψ(n) < ln n, it was derived from inequality () that which was also rewritten as n!(n + ) s- (s + )(s + ) · · · (s + n -) ≤ ( + s) ≤ (n -)!n s (s + )(s + ) · · · (s + n -) () and so a simple proof of Euler's product formula in the segment  ≤ s ≤  was shown by letting n → ∞ in (). This suggests us the following double inequality:

Remark  Double inequalities () and () can be further rearranged as
for real numbers s, t and x ∈ (-min{s, t}, ∞), where α(x) ∼ x and β(x) ∼ x as x → ∞. For detailed information on the type of inequalities like (), please refer to [] and related references therein.
Remark  Inequality () can be rewritten as  ≤ (n + ) (n + s) for n ∈ N and  ≤ s ≤ .
Remark  In the reviews on the paper [] by the Mathematical Reviews and the Zentralblatt MATH, there is not a word to comment on inequalities in () and (). However, these two double inequalities later became a major source of a series of studies on bounding the ratio of two gamma functions.

Chu's double inequality
In , by discussing that Remark  After letting n = k + , inequality () becomes which is the same as (). Taking n = k +  in () leads to inequalities () and ( for n ∈ N. Therefore, Chu discussed equivalently the necessary and sufficient conditions such that the sequence B c (n) for n ∈ N is monotonic.
Recently, necessary and sufficient conditions for the general function

Lazarević-Lupaş's claim
In , among other things, the function on (, ∞) for α ∈ (, ) was claimed in [, Theorem ] to be decreasing and convex, and so

Kershaw's first double inequality
In , motivated by inequality () obtained in [], among other things, Kershaw presented in [] the following double inequality: for  < s <  and x > . In the literature, it is called Kershaw's first double inequality for the ratio of two gamma functions.
Kershaw's proof for () Define the function g β by for x >  and  < s < , where the parameter β is to be determined. It is not difficult to show, with the aid of Wendel's limit (), that To prove double inequality (), define from which it follows that This leads to  , then G strictly decreases, and since G(x) →  as x → ∞, it follows that G(x) >  for x > . However, from (), this implies that g β (x) > g β (x + ) for x > , and so g β (x) > g β (x + n). Take the limit as n → ∞ to give the result that g β (x) > , which can be rewritten as the left-hand side inequality in (). The corresponding upper bound can be verified by a similar argument when β = -  + (s +   ) / , the only difference being that in this case g β strictly increases to unity.

Remark 
The spirit of Kershaw's proof is similar to Chu's in [, Theorem ], as shown by (). This idea or method was also utilized independently in [-] to construct, for various purposes, a number of inequalities of the type for s >  and real number x ≥ . http://www.journalofinequalitiesandapplications.com/content/2013/1/542

Remark  It is easy to see that inequality () refines and extends inequalities () and ().
Remark  Inequality () may be rearranged as for x >  and  < s < . Theorem  The function z s,t (x) is either convex and decreasing for |t -s| <  or concave and increasing for |t -s| > .

Remark  Direct computation yields
To prove the positivity of function (), the following formula and inequality are used as basic tools in the proof of [, Theorem ]. . For x > -, Remark  As consequences of Theorem , the following useful conclusions are derived. . The function is decreasing and convex from (-t, ∞) onto (t -  , t), where t ∈ R. http://www.journalofinequalitiesandapplications.com/content/2013/1/542 . For all x > , . For all x >  and t > , holds if |t -s| <  and reverses if |t -s| > . Remark  It is clear that double inequality () can be deduced directly from the decreasingly monotonic property of (). Furthermore, from the decreasingly monotonic and convex properties of () on (-t, ∞), inequality () and

Recent advances
Finally, we would like to state some new results related to or originating from Elezović-Giordano-Pečarić's Theorem  above.

Alternative proofs of Elezović-Giordano-Pečarić's theorem
The key step of verifying Theorem  is to prove the positivity of the right-hand side in (), which involves divided differences of the digamma and trigamma functions. The biggest barrier or difficulty to prove the positivity of () is mainly how to deal with the squared term in ().

Chen's proof
In [], the barrier mentioned above was overcome by virtue of the well-known convolution theorem [] for Laplace transforms, and so Theorem  for the special case s +  > t > s ≥  was proved. Perhaps this is the first try to provide an alternative of Theorem , although it was partially successful formally. In [, ], by making use of the convolution theorem for the Laplace transform and the logarithmically convex properties of the function q α,β (x) on (, ∞), an alternative proof of Theorem  was supplied.

Guo-Qi's first proof
In [, ], by considering monotonic properties of the function and still employing the convolution theorem for the Laplace transform, Theorem  was completely verified again.
Remark  For more information on the function q α,β (t) and its applications, please refer to [, , -] and related references therein.

Guo-Qi's second proof
In [-], the complete monotonic properties of the function on the right-hand side of () were established as follows.
Since the complete monotonicity of the functions s,t (x) ands,t (x) mean the positivity and negativity of the function s,t (x), an alternative proof of Theorem  was provided once again.
One of the key tools or ideas used in the proofs of Theorem  is the following simple but specially successful conclusion: If f (x) is a function defined on an infinite interval I ⊆ R and it satisfies lim x→∞ f (x) = δ and f (x)f (x + ε) >  for x ∈ I and some fixed number ε > , then f (x) > δ on I.
It is clear that Theorem  is a generalization of inequality ().

Complete monotonicity of divided differences
In order to prove Theorem , the following complete monotonic properties of a function related to a divided difference of the psi function were discovered in [], the preprint of []. To the best of our knowledge, the complete monotonicity of functions involving divided differences of the psi and polygamma functions were investigated first in [-].

Inequalities for sums
As consequences of proving Theorem  along different approach from [] and its preprint [], the following algebraic inequalities for sums were procured in [, ] accidentally.
Theorem  Let k be a nonnegative integer and let θ >  be a constant.
If a >  and b > , then holds for ba > -θ and reverses for ba < -θ .
If a < -θ and b > , then inequality () holds and inequality () is valid for a + b + θ >  and is reversed for a + b + θ < .
If a >  and b < -θ , then inequality () is reversed and inequality () holds for a+b+θ <  and reverses for a + b + θ > .
Moreover, the following equivalent relation between inequality () and Theorem  was found in [, ].
Theorem  Inequality () for positive numbers a and b is equivalent to Theorem .

Recent advances
Recently, some applications, extensions, and generalizations of Theorems  to , and related conclusions have been investigated in several recently or immediately published manuscripts such as [-]. For example, Theorem  stated in Remark  was obtained in [].
The complete monotonicity of the q-analogue of the function δ , defined by () was researched in [, ]. http://www.journalofinequalitiesandapplications.com/content/2013/1/542 Remark  This article is a slightly revised version of the preprint [] and a companion paper of the preprint [] and the articles [, ] whose preprints are [, ], respectively.