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Qi and Luo Journal of Inequalities and Applications 2013, 2013:542http://www.journalofinequalitiesandapplications.com/content/2013/1/542

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Bounds for the ratio of two gamma functions:fromWendel’s asymptotic relation toElezovic-Giordano-Pecaric’s theoremFeng Qi1,2* and Qiu-Ming Luo3

*Correspondence:[email protected];[email protected];[email protected] of Mathematics, HenanPolytechnic University, Jiaozuo City,Henan Province 028043, China2College of Mathematics, InnerMongolia University forNationalities, Tongliao City, InnerMongolia Autonomous Region028043, ChinaFull list of author information isavailable at the end of the article

AbstractIn the expository review and survey paper dealing with bounds for the ratio of twogamma functions, along one of the main lines of bounding the ratio of two gammafunctions, the authors look back and analyze some known results, including Wendel’sasymptotic relation, Gurland’s, Kazarinoff’s, Gautschi’s, Watson’s, Chu’s, Kershaw’s, andElezovic-Giordano-Pecaric’s inequalities, Lazarevic-Lupas’s claim, and other monotonicand convex properties. On the other hand, the authors introduce some relatedadvances on the topic of bounding the ratio of two gamma functions in recent years.MSC: Primary 33B15; secondary 26A48; 26A51; 26D07; 26D15; 44A10

Keywords: bound; inequality; ratio of two gamma functions; divided difference; psifunction; polygamma function; completely monotonic function; Wallis’ formula;inequality for sums

1 IntroductionRecall [, ChapterXIII] and [, Chapter IV] that a function f is said to be completelymono-tonic on an interval I if f has derivatives of all orders on I and

(–)nf (n)(x) ≥ ()

for x ∈ I and n ≥ . The celebrated Bernstein-Widder theorem [, p., Theorem a]states that a function f is completely monotonic on [,∞) if and only if

f (x) =∫ ∞

e–xs dμ(s), ()

where μ is a nonnegative measure on [,∞) such that the integral () converges for allx > . This tells us that a completelymonotonic function f on [,∞) is a Laplace transformof the measure μ.It is well known that the classical Euler gamma function may be defined for x > by

�(x) =∫ ∞

tx–e–t dt. ()

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-

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edge that the special functions �(x), ψ(x), and ψ (k)(x) for k ∈ N are fundamental and im-portant 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 bound-ing the ratio of two gamma functions are various, including establishment of asymptoticrelation, refinements of Wallis’ formula, approximation to π , needs in statistics and othermathematical 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 Wen-del’s asymptotic relation, Gurland’s approximation to π , Kazarinoff’s refinement ofWallis’formula, Gautschi’s double inequality, Watson’s monotonicity, Chu’s refinement of Wal-lis’ formula, Lazarević-Lupaş’s claim onmonotonic and convex properties, Kershaw’s firstdouble 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 andpolygamma functions, inequalities for sums and related results.

2 Wendel’s asymptotic relationOur starting point is the paper published in by Wendel, which is the earliest relatedone we could search out to the best of our ability.In order to establish the classical asymptotic relation

limx→∞

�(x + s)xs�(x)

= ()

for real s and x, by using Hölder’s inequality for integrals, Wendel [] proved elegantly thedouble inequality

(x

x + s

)–s

≤ �(x + s)xs�(x)

≤ ()

for < s < and x > .

Wendel’s original proof Let

< s < ,p =s, q =

pp –

=

– s,

f (t) = e–sttsx, g(t) = e–(–s)tt(–s)x+s–,

and apply Hölder’s inequality for integrals and the recurrence formula

�(x + ) = x�(x) ()

for x > to obtain

�(x + s) =∫ ∞

e–ttx+s– dt ≤

(∫ ∞

e–ttx dt

)s(∫ ∞

e–ttx– dt

)–s

=[�(x + )

]s[�(x)

]–s = xs�(x). ()



Replacing s by – s in (), we get

�(x + – s) ≤ x–s�(x), ()

from which we obtain

�(x + ) ≤ (x + s)–s�(x + s) ()

by substituting x + s for x.Combining () and (), we get

x(x + s)–s

�(x)≤ �(x + s)≤ xs�(x).

Therefore, inequality () follows.Letting x tend to infinity in () yields () for < s < . The extension to all real s is im-

mediate on repeated application of (). �

Remark Inequality () can be rewritten for < s < and x > as

x–s ≤ �(x + )�(x + s)

≤ (x + s)–s ()

or

≤[

�(x + )�(x + s)

]/(–s)

– x ≤ s. ()

Remark Using recurrence formula () and double inequality () repeatedly yields

x–s∏m

i=(x + i)∏n–i= (x + i + s)

≤ �(x +m + )�(x + n + s)

≤ (x + s)–s∏m

i=(x + i)∏n–i= (x + i + s)

()

for x > and < s < , where m and n are positive integers. This implies that basing onrecurrence formula () and double inequality (), one can bound the ratio �(x+a)

�(x+b) for anypositive numbers x, a and b. Conversely, double inequality () reveals that one can alsodeduce corresponding bounds of the ratio �(x+)

�(x+s) for x > and < s < from bounds of theratio �(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,

limx→∞

[xb–a

�(x + a)�(x + b)

]= . ()

Limits () and () are equivalent to each other since

xt–s�(x + s)�(x + t)

=�(x + s)xs�(x)

· xt�(x)�(x + t)

.

Hence, limit () is called Wendel’s asymptotic relation in the literature.



Remark Double inequality () or () is more meaningful than limit () or (), sincethe former implies the latter, but not conversely.

Remark For more general results about the asymptotic expansion of a ratio of twogamma functions, please refer to [, ].

Remark Due to unknown reasons, Wendel’s paper [] and double inequality () or ()were seemingly neglected by nearly all mathematicians for more than fifty years until itwas mentioned in [], to the best of our knowledge.

Remark There has been a lot of literature about asymptotic expansions of a ratio oftwo gamma functions. Because this expository review and survey paper is devoted to in-equalities and complete monotonicity, we will not further survey the topic of asymptoticexpansions of a quotient of two gamma functions. For more information, please refer to[, –] and closely related references therein.

3 Gurland’s double inequalityBy making use of a basic theorem in mathematical statistics concerning unbiased estima-tors with minimum variance, Gurland [] presented the following inequality:

[�((n + )/)

�(n/)

]

<n

n + ()

for n ∈ N, and so taking respectively n = k and n = k + for k ∈ N in () yields a closerapproximation to π :

k + (k + )

[(k)!!

(k – )!!

]

< π <

k +

[(k)!!

(k – )!!

]

, k ∈N. ()

Remark Taking respectively n = k and n = k – for k ∈N in () leads to

√k +

<

�(k + )�(k + /)

<k√k –

=k√

k – /, k ∈N. ()

This is better than double inequality () for x = k and s = .

Remark Double inequality () may be rearranged as

√k +

<

�(k + )�(k + /)

<k + √k +

=k + /√

k + / + /, k ∈N. ()

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 beobtained only if letting m ∈ N in n = m – in (). However, this is an illusion, since the



lower bound of the following double inequality:

k

k + ·√(k +m) – k +m –

m–∏i=

[ +

(k + i)

]

<�(k + )

�(k + /)<

(k

k –

) (k +m) – √(k +m) –

m–∏i=

[ –

(k + i)

], ()

which is derived from taking respectively n = (k + m – ) and n = (k + m – ) – fork ∈ 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 thecorresponding one in ()?

Remark The left-hand side inequality in () or () may be rearranged as

<

[�(k + )

�(k + /)

]

– k <+

(k + )

, k ∈N. ()

From this, it is easier to see that inequality () refines double inequality () for x = k ands =

.

Remark It is noted that inequality () was recovered in [] and extended in [] bydifferent approaches respectively. See Section and Section below.

Remark Just like the paper [], Gurland’s paper [] was ignored except that it wasmentioned in [, ]. The famous monograph [] recorded neither of the papers [, ].

4 Kazarinoff’s double inequalityStarting from one form of the celebrated formula of John Wallis,

√π (n + /)

<(n – )!!(n)!!

<√πn

, n ∈ N, ()

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

(n – )!!(n)!!

=√

π [n + θ (n)]()

satisfies < θ (n) <

for n ∈N. This implies

√π (n + /)

<(n – )!!(n)!!

<√

π (n + /), n ∈N. ()

Remark It was said in [] that it is unquestionable that inequalities similar to () canbe improved indefinitely but at a sacrifice of simplicity, which is why inequality () hadsurvived so long.



Remark Kazarinoff’s proof of () is based upon the property

[lnφ(t)

]′′ –{[lnφ(t)

]′} > ()

of the function

φ(t) =∫ π/

sint xdx

=√

π

· �((t + )/)�((t + )/)

()

for – < t < ∞. Inequality () was proved by making use of well-known Legendre’s for-mula

ψ(x) = –γ +∫

tx– – t –

dt ()

for x > and estimating the integrals

∫

xt

+ xdx and

∫

xt lnx + x

dx. ()

Since () is equivalent to the statement that the reciprocal of φ(t) has an everywherenegative second derivative, therefore, for any positive t, φ(t) is less than the harmonicmean of φ(t – ) and φ(t + ), which implies

�((t + )/)�((t + )/)

<√t +

, t > –. ()

As a subcase of this result, the right-hand side inequality in () is established.

Remark Using recurrence formula () in () gives

[�((t + )/)

�(t/)

]

<t

t + ()

for t > , which extends inequality (). This shows that Kazarinoff’s paper [] containsmore general conclusions and that all results in [] stated in paper [] contain manygeneral conclusions, and that all results in [] are stated below.Replacing t by t in () or () and rearranging yield

�(t + )�(t + /)

>√t +

⇐⇒[

�(t + )�(t + /)

]

– t >

()

for t > , which extends the left-hand side inequality in () and (). Replacing t by t – in () or () produces

�(t + )�(t + /)

<t√t –

()



for t > , which extends the right-hand side inequality in (). Replacing t by t + in ()

or () and rearranging gives

�(t + )�(t + /)

<t + √t +

()

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

θ (n) =π

[(n)!!

(n – )!!

]

– n =[

�(n + )�(n + /)

]

– n ()

for n ∈N. Then inequality () is equivalent to

<

[�(n + )

�(n + /)

]

– n <, n ∈ N. ()

Remark Inequality () may be rewritten as

ψ ′(t +

)–ψ ′

(t +

)>

[ψ

(t +

)–ψ

(t +

)]

()

for t > –. Letting u = t+ in the above inequality yields

ψ ′(u) –ψ ′(u +

)>

[ψ(u) –ψ

(u +

)]

()

for u > . This inequality has been generalized in [] to the complete monotonicity of afunction involving divided differences of the digamma and trigamma functions as follows.

Theorem [] For real numbers s, t, α =min{s, t}, and λ, let

s,t;λ(x) =

⎧⎨⎩[ψ(x+t)–ψ(x+s)

t–s ] + λψ ′(x+t)–ψ ′(x+s)

t–s , s �= t,

[ψ ′(x + s)] + λψ ′′(x + s), s = t()

on (–α,∞). Then the function s,t;λ(x) has the following complete monotonicity:. For < |t – s| < ,

(a) the function s,t;λ(x) is completely monotonic on (–α,∞) if and only if λ ≤ ,(b) so is the function –s,t;λ(x) if and only if λ ≥

|t–s| ;. For |t – s| > ,

(a) the function s,t;λ(x) is completely monotonic on (–α,∞) if and only if λ ≤ |t–s| ,

(b) so is the function –s,t;λ(x) if and only if λ ≥ ;. For s = t, the function s,s;λ(x) is completely monotonic on (–s,∞) if and only if λ ≤ ;. For |t – s| = ,

(a) the function s,t;λ(x) is completely monotonic if and only if λ < ,(b) so is the function –s,t;λ(x) if and only if λ > ,(c) and s,t;(x)≡ .



Remark Taking in Theorem λ = s– t > produces that the function �(x+s)�(x+t) on (–t,∞)

is increasingly convex if s – t > and increasingly concave if < s – t < .

5 Watson’s monotonicityIn , motivated by the result in [] mentioned in Section ,Watson [] observed that

x

· [�(x + )]

[�(x + /)]= F

(–,–

;x;

)

= +x

+

x(x + )+

∞∑r=

[(–/) · (/) · (/) · (r – /)]

r!x(x + ) · · · (x + r – )()

for x > – , which implies that the more general function

θ (x) =[

�(x + )�(x + /)

]

– x ()

for x > – , whose special case is the sequence θ (n) for n ∈ N defined in () or (), is

decreasing and

limx→∞ θ (x) =

and limx→(–/)+

θ (x) =. ()

This apparently implies the sharp inequalities

< θ (x) <

()

for x > – ,

√x +

<

�(x + )�(x + /)

≤√x +

+

[�(/)�(/)

]

=√x + . . . . ()

for x ≥ – , and, by Wallis’ cosine formula [],

√π (n + /π – )

≤ (n – )!!(n)!!

<√

π (n + /), n ∈N. ()

In [], an alternative proof of double inequality () was also provided as follows. Let

f (x) =√π

∫ π/

cosx t dt =

√π

∫ ∞

exp

(–xt

) t exp(–t/)√ – exp(–t)

dt ()

for x > . By using the fairly obvious inequalities

√ – exp

(–t

) ≤ t ()

and

t exp(–t/)√ – exp(–t)

=t√

sinh(t/)≤ , ()



we have, for x > – ,

√π

∫ ∞

exp

(–(x + /)t

)dt < f (x) <

√π

∫ ∞

exp

(–(x + /)t

)dt,

that is to say,

√x + /

< f (x) <√

x + /. ()

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 ≥ .

Remark Formula () implies the complete monotonicity of the function θ (x) definedby () on (–

,∞).

6 Gautschi’s double inequalitiesThe main aim of the paper [] was to establish the double inequality

(xp + )/p – x

< exp∫ ∞

xe–t

p dt ≤ cp[(

xp +cp

)/p

– x]

()

for x ≥ and p > , where

cp =[�

( +

p

)]p/(p–)

()

or cp = .By an easy transformation, inequality () was written in terms of the complementary

gamma function

�(a,x) =∫ ∞

xe–tta– dt ()

as

p[(x + )/p – x/p]

< ex�(p,x

)≤ pcp

[(x +

cp

)/p

– x/p]

()

for x ≥ and p > . In particular, letting p→ ∞, the double inequality

ln

( +

x

)≤ exE(x)≤ ln

( +

x

)()

for the exponential integral E(x) = �(,x) for x > was derived from (), in which thebounds exhibit the logarithmic singularity of E(x) at x = .



As a direct consequence of inequality () for p = s , x = and cp = , the following simple

inequality for the gamma function was deduced:

s– ≤ �( + s) ≤ , ≤ s ≤ . ()

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

e(s–)ψ(n+) ≤ �(n + s)�(n + )

≤ ns– ()

for ≤ s ≤ and n ∈N than () by proving that the function

f (s) =

– sln

�(n + s)�(n + )

()

is monotonically decreasing for ≤ s < . Since ψ(n) < lnn, it was derived from inequality() that

(

n +

)–s

≤ �(n + s)�(n + )

≤(n

)–s

, ≤ s ≤ , ()

which was also rewritten as

n!(n + )s–

(s + )(s + ) · · · (s + n – )≤ �( + s)≤ (n – )!ns

(s + )(s + ) · · · (s + n – )()

and so a simple proof of Euler’s product formula in the segment ≤ s ≤ was shown byletting n → ∞ in ().

Remark Double inequalities () and () can be further rearranged as

n–s ≤ �(n + )�(n + s)

≤ exp(( – s)ψ(n + )

)()

and

n–s ≤ �(n + )�(n + s)

≤ (n + )–s ()

for n ∈N and ≤ s ≤ .

Remark The upper bounds in () and () have the following relationship:

(n + s)–s ≤ exp(( – s)ψ(n + )

)()

for ≤ s ≤ and n ∈ N, and inequality () reverses for s > e–γ – = . . . . , since

the function

Q(x) = eψ(x+) – x ()



was proved in [, Theorem ] to be strictly decreasing on (–,∞), with

limx→∞Q(x) =

. ()

Thismeans thatWendel’s double inequality () andGautschi’s first double inequality ()are not included in each other but they all contain Gautschi’s second double inequality().

Remark The right-hand side inequality in () may be rearranged as

[�(n + )�(n + s)

]/(–s)

≤ exp(ψ(n + )

), n ∈N. ()

This suggests us the following double inequality:

exp(ψ

(α(x)

))<

[�(x + t)�(x + s)

]/(t–s)

≤ exp(ψ

(β(x)

))()

for real numbers s, t and x ∈ (–min{s, t},∞), where α(x) ∼ x and β(x) ∼ x as x → ∞. Fordetailed information on the type of inequalities like (), please refer to [] and relatedreferences therein.

Remark Inequality () can be rewritten as

≤[

�(n + )�(n + s)

]/(–s)

– n≤ ()

for n ∈N and ≤ s ≤ .

Remark In the reviews on the paper [] by the Mathematical Reviews and the Zen-tralblattMATH, 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 bound-ing the ratio of two gamma functions.

7 Chu’s double inequalityIn , by discussing that

bn+(c)� bn(c) ()

if and only if ( – c)n + – c� , where

bn(c) =(n – )!!(n)!!

√n + c, ()

it was demonstrated in [, Theorem ] that

√π [n + (n + )/(n + )]

<(n – )!!(n)!!

<√

π (n + /), n ∈N. ()



As an application of (), by using �( ) =√

π and (), the following double inequality

√n –

<

�(n/)�(n/ – /)

≤√(n – )

n – ()

for positive integers n≥ was given in [, Theorem ].

Remark After letting n = k + , inequality () becomes

√k –

<

�(k + /)�(k)

<k√

k + /, ()

which is the same as (). Taking n = k + in () leads to inequalities () and ().Notice that the reasoning directions in the two papers [, ] are opposite:

(n – )!!(n)!!

[]�⇒⇐�[]

�(n/)�(n/ – /)

. ()

To some extent, the results obtained by Gurland in [] and by Chu in [] are equivalentto each other, and they are all special cases of those obtained by Kazarinoff in [].

Remark By Wallis’ cosine formula [], sequence () may be rewritten as

bn(c) =√π

· �(n + /)�(n + )

√n + c� √

πBc(n) ()

for n ∈ N. Therefore, Chu discussed equivalently the necessary and sufficient conditionssuch that the sequence Bc(n) for n ∈N is monotonic.Recently, necessary and sufficient conditions for the general function

Ha,b,c(x) = (x + c)b–a�(x + a)�(x + b)

()

on (–ρ,∞), where a, b, and c are real numbers and ρ =min{a,b, c}, to be logarithmicallycompletely monotonic are presented in [, ]. A positive function f is said to be loga-rithmically completely monotonic on an interval I ⊆R if it has derivatives of all orders onI and its logarithm ln f satisfies (–)k[ln f (x)](k) ≥ for k ∈N on I , see [–].

8 Lazarevic-Lupas’s claimIn , among other things, the function

θα(x) =[

�(x + )�(x + α)

]/(–α)

– x ()

on (,∞) for α ∈ (, ) was claimed in [, Theorem ] to be decreasing and convex, andso

α

<

[�(x + )�(x + α)

]/(–α)

– x ≤ [�(α)

]/(–α). ()



Remark The proof of [, Theorem ] is wrong, see [, Remark .] and [, p.].However, the statements in [, Theorem ] are correct.

9 Kershaw’s first double inequalityIn , motivated by inequality () obtained in [], among other things, Kershaw pre-sented in [] the following double inequality:

(x +

s

)–s

<�(x + )�(x + s)

<[x –

+

(s +

)/]–s

()

for < s < and x > . In the literature, it is called Kershaw’s first double inequality for theratio of two gamma functions.

Kershaw’s proof for () Define the function gβ by

gβ (x) =�(x + )�(x + s)

(x + β)s– ()

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

limx→∞ gβ (x) = . ()

To prove double inequality (), define

G(x) =gβ (x)

gβ (x + )=x + sx +

(x + β + x + β

)–s

, ()

from which it follows that

G′(x)G(x)

=( – s)[(β + β – s) + (β – s)x](x + )(x + s)(x + β)(x + β + )

.

This leads to. if β = s

, then G′(x) < for x > ;. if β = –

+ (s + )

/, then G′(x) > for x > .Consequently, ifβ = s

, thenG strictly decreases, and sinceG(x) → as x→ ∞, it followsthat G(x) > for x > . However, from (), this implies that gβ (x) > gβ (x+ ) for x > , andso gβ (x) > gβ (x+ n). Take the limit as n→ ∞ to give the result that gβ (x) > , which can berewritten as the left-hand side inequality in (). The corresponding upper bound can beverified by a similar argument when β = –

+ (s + )

/, the only difference being that inthis case gβ strictly increases to unity. �

Remark The spirit of Kershaw’s proof is similar to Chu’s in [, Theorem ], as shownby (). This idea or method was also utilized independently in [–] to construct, forvarious purposes, a number of inequalities of the type

(x + α)s– <�(x + s)�(x + )

< (x + β)s– ()

for s > and real number x ≥ .



Remark It is easy to see that inequality () refines and extends inequalities () and().

Remark Inequality () may be rearranged as

s<

[�(x + )�(x + s)

]/(–s)

– x <(s +

)/

–

()

for x > and < s < .

10 Elezovic-Giordano-Pecaric’s theoremInequalities (), (), (), and (), sequence (), and functions () and () stronglysuggest us to consider the monotonic and convex properties of the general function

zs,t(x) =

⎧⎨⎩[�(x+t)

�(x+s) ]/(t–s) – x, s �= t,

eψ(x+s) – x, s = t()

for x ∈ (–α,∞), where s and t are two real numbers and α =min{s, t}.In , Elezović, Giordano, and Pečarić gave in [, Theorem ] a perfect solution to

the monotonic and convex properties of the function zs,t(x) as follows.

Theorem The function zs,t(x) is either convex and decreasing for |t – s| < or concaveand increasing for |t – s| > .

Remark Direct computation yields

z′′s,t(x)

zs,t(x) + x=

[ψ(x + t) –ψ(x + s)

t – s

]

+ψ ′(x + t) –ψ ′(x + s)

t – s. ()

To prove the positivity of function (), the following formula and inequality are used asbasic tools in the proof of [, Theorem ].. For x > –,

ψ(x + ) = –γ +∞∑k=

(k–

x + k

). ()

. If a ≤ b < c≤ d, then

ab

+cd

>ac

+bd

. ()

Remark As consequences of Theorem, the following useful conclusions are derived.. The function

eψ(x+t) – x ()

is decreasing and convex from (–t,∞) onto (t – , t), where t ∈R.



. For all x > ,

eψ(x) <

ψ ′(x)or ψ(x) < – lnψ ′(x). ()

. For all x > and t > ,

ln

(x + t –

)< ψ(x + t) < ln

(x + eψ(t)). ()

. For x > –α, the inequality

[�(x + t)�(x + s)

]/(t–s)

<t – s

ψ(x + t) –ψ(x + s)()

holds if |t – s| < and reverses if |t – s| > .

Remark In fact, function () is deceasing and convex on (–t,∞) for all t ∈R. See [,Theorem ].

Remark It is clear that double inequality () can be deduced directly from the de-creasingly monotonic property of (). Furthermore, from the decreasingly monotonicand convex properties of () on (–t,∞), inequality () and

ψ ′′(x) +[ψ ′(x)

] > ()

on (,∞) can be derived straightforwardly.

11 Recent advancesFinally, we would like to state some new results related to or originating from Elezović-Giordano-Pečarić’s Theorem above.

Alternative proofs of Elezovic-Giordano-Pecaric’s theoremThe 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 biggestbarrier or difficulty to prove the positivity of () is mainly how to deal with the squaredterm in ().

Chen’s proofIn [], the barrier mentioned above was overcome by virtue of the well-known con-volution theorem [] for Laplace transforms, and so Theorem for the special cases + > t > s ≥ was proved. Perhaps this is the first try to provide an alternative of Theo-rem , although it was partially successful formally.

Qi-Guo-Chen’s proofFor real numbers α and β with (α,β) /∈ {(, ), (, )} and α �= β , let

qα,β (t) =

⎧⎨⎩

e–αt–e–βt

–e–t , t �= ,

β – α, t = .()



In [, ], by making use of the convolution theorem for the Laplace transform and thelogarithmically convex properties of the function qα,β (x) on (,∞), an alternative proof ofTheorem was supplied.

Guo-Qi’s first proofIn [, ], by considering monotonic properties of the function

Qs,t;λ(u) = qα,β (u)qα,β (λ – u), λ ∈R ()

and still employing the convolution theorem for the Laplace transform, Theorem wascompletely verified again.

Remark Formore information on the function qα,β (t) and its applications, please referto [, , –] and related references therein.

Guo-Qi’s second proofIn [–], the complete monotonic properties of the function on the right-hand side of() were established as follows.

Theorem Let s and t be two real numbers and α =min{s, t}. Define

s,t(x) =

⎧⎨⎩[ψ(x+t)–ψ(x+s)

t–s ] + ψ ′(x+t)–ψ ′(x+s)t–s , s �= t,

[ψ ′(x + s)] +ψ ′′(x + s), s = t()

on x ∈ (–α,∞). Then the functions s,t(x) for |t – s| < and –s,t(x) for |t – s| > arecompletely monotonic on x ∈ (–α,∞).

Since the complete monotonicity of the functions s,t(x) and –s,t(x) mean the positiv-ity and negativity of the function s,t(x), an alternative proof of Theorem was providedonce 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 limx→∞ f (x) = δ and f (x) – f (x + ε) > for x ∈ I and some fixed numberε > , then f (x) > δ on I .It is clear that Theorem is a generalization of inequality ().

Complete monotonicity of divided differencesIn order to prove Theorem , the following complete monotonic properties of a functionrelated to a divided difference of the psi function were discovered in [], the preprint of[].

Theorem Let s and t be two real numbers and α =min{s, t}. Define

δs,t(x) =

⎧⎨⎩

ψ(x+t)–ψ(x+s)t–s – x+s+t+

(x+s)(x+t) , s �= t,

ψ ′(x + s) – x+s –

(x+s) , s = t

()

on x ∈ (–α,∞). Then the functions δs,t(x) for |t – s| < and –δs,t(x) for |t – s| > are com-pletely monotonic on x ∈ (–α,∞).



To the best of our knowledge, the complete monotonicity of functions involving divideddifferences of the psi and polygamma functions were investigated first in [–].

Inequalities for sumsAs consequences of provingTheorem 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

k∑i=

(a + θ )i+(b + θ )k–i+

+k∑i=

ai+bk–i+

> k∑i=

(a + θ )i+bk–i+

()

holds for b – a > –θ and reverses for b – a < –θ .If a < –θ and b < –θ , then the inequalities

k∑i=

(a + θ )i+(b + θ )k–i+

+k∑i=

ai+bk–i+

> k∑i=

(a + θ )i+bk–i+

()

and

k+∑i=

(a + θ )i+(b + θ )k–i+

+k+∑i=

ai+bk–i+

< k+∑i=

(a + θ )i+bk–i+

()

hold for b – a > –θ and reverse for b – a < –θ .If –θ < a < and –θ < b < , then inequality () holds and inequality () is valid for

a + b + θ > and is reversed for a + b + 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 > .If b = a – θ , then inequalities (), () and () become equalities.

Moreover, the following equivalent relation between inequality () and Theorem wasfound in [, ].

Theorem Inequality () for positive numbers a and b is equivalent to Theorem .

Recent advancesRecently, some applications, extensions, and generalizations of Theorems to , and re-lated conclusions have been investigated in several recently or immediately publishedmanuscripts such as [–]. For example, Theorem stated in Remark was obtainedin [].The complete monotonicity of the q-analogue of the function δ, defined by () was

researched in [, ].



Remark This article is a slightly revised version of the preprint [] and a companionpaper of the preprint [] and the articles [, ] whose preprints are [, ], respec-tively.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally to the manuscript and read and approved the final manuscript.

Author details1Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province 028043, China. 2College ofMathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region 028043,China. 3Department of Mathematics, Chongqing Normal University, Chongqing City, 401331, China.

AcknowledgementsThe original version of this article was reported on Thursday 24 July 2008 as a talk in the seminar held at the ResearchGroup in Mathematical Inequalities and Applications (RGMIA), School of Computer Science and Mathematics, VictoriaUniversity, Australia, while the first author was visiting the RGMIA between March 2008 and February 2009 by the grantfrom the China Scholarship Council. The first author would like to express many thanks to Professors Pietro Cerone andServer S. Dragomir and other local colleagues for their invitation and hospitality throughout this period. This work waspartially supported by the NNSF of China under Grant No. 11361038, by the NSF Project of Chongqing City under GrantNo. CSTC2011JJA00024, by the Research Project of Science and Technology of Chongqing Education Commission underGrant 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 theoriginal version of this manuscript.

Received: 27 August 2013 Accepted: 22 October 2013 Published: 19 Nov 2013

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