A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

Abstract

A class of functions involving the divided differences of the psi function
and the polygamma functions and originating from Kershaw's double inequality
are proved to be completely monotonic. As applications of these results, the
monotonicity and convexity of a function involving ratio of two gamma functions
and originating from establishment of the best upper and lower bounds in
Kershaw's double inequality are derived, two sharp double inequalities
involving ratios of double factorials are recovered, the probability integral
or error function is estimated, a double inequality for ratio of the volumes of
the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is
deduced, and a symmetrical upper and lower bounds for the gamma function in
terms of the psi function is generalized.

Full text

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arXiv:0903.1430v1 [math.CA] 8 Mar 2009
A CLASS OF COMPLETELY MONOTONIC FUNCTIONS
INVOLVING DIVIDED DIFFERENCES OF THE PSI AND
POLYGAMMA FUNCTIONS AND SOME APPLICATIONS
FENG QI AND BAI-NI GUO
Abstract. A class of functions involving the divided differences of the psi
function and the polygamma functions and originating from Kershaw’s double
inequality are proved to be completely monotonic. As applications of these
results, the monotonicity and convexity of a function involving ratio of two
gamma functions and originating from establishment of the best upper and
lower bounds in Kershaw’s double inequality are derived, two sharp double
inequalities involving ratios of double factorials are recovered, the probability
integral or error function is estimated, a double inequality for ratio of the
volumes of the unit balls in Rn−1and Rnrespectively is deduced, and a
symmetrical upper and lower bounds for the gamma function in terms of the
psi function is generalized.
1. Introduction
Recall [29, Chapter XIII] and [57, 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
(−1)nf(n)(x) ≥ 0(1)
for x ∈ I and n ≥ 0. The famous Bernstein’s Theorem in [57, p. 160, Theorem 12a]
states that a function f is completely monotonic on [0,∞) if and only if
?∞
where µ is a nonnegative measure on [0,∞) such that the integral (2) converges for
all x > 0. This expresses that a completely monotonic function f on [0,∞) is a
Laplace transform of the measure µ.
Recall also [5, 46, 49] that a positive function f is called logarithmically com-
pletely monotonic on an interval I if f has derivatives of all orders on I and its
logarithm lnf satisfies
(−1)k[lnf(x)](k)≥ 0
for all k ∈ N on I. It was proved explicitly in [10, 42, 45, 46, 53] by different
approaches that any logarithmically completely monotonic function must be com-
pletely monotonic, but not conversely. It was pointed out in [10, Theorem 1.1]
and [24, 50] that the logarithmically completely monotonic functions on [0,∞) are
those completely monotonic functions on [0,∞) for which the representing measure
µ in (2) is infinitely divisible in the convolution sense: For each n ∈ N there exists
a positive measure ν on [0,∞) with n-th convolution power equal to µ.
f(x) =
0
e−xsdµ(s), (2)
(3)
2000 Mathematics Subject Classification. 05A10, 26D15, 26A48, 26A51, 33B15, 33B20, 65R10.
Key words and phrases. completely monotonic function, logarithmically completely monotonic
function, divided difference, psi function, polygamma function, Kershaw’s inequality, probability
integral, error function, double factorial, ratio of the volumes of the unit balls in Rn, monotonicity,
convexity, inequality, generalization, application.
This paper was typeset using AMS-LATEX.
1

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2 F. QI AND B.-N. GUO
It is well-known that the classical Euler gamma function
Γ(x) =
?∞
0
tx−1e−tdt (4)
for x > 0, its logarithmic derivative, denoted by ψ(x) =Γ′(x)
functions ψ(i)(x) for i ∈ N are several of the most important special functions and
have much extensive applications in many branches such as statistics, probabil-
ity, number theory, theory of 0-1 matrices, graph theory, combinatorics, physics,
engineering, and other mathematical sciences.
The ratioΓ(x+p)
Γ(x+q)for x+p > 0 and x+q > 0 of two gamma functions, called Wallis
function or ratio in the literature, has been investigated since 1948 in [56] at least.
Now there exist a lot of conclusions on Wallis ratio, its variants, generalizations
and applications, for example, [2, 11, 21, 23, 24, 28, 30, 31, 32, 33, 34, 35, 40, 41,
44, 47, 48] and related references therein.
In [26], D. Kershaw proved a double inequality
Γ(x), and the polygamma
?
x +s
2
?1−s
<Γ(x + 1)
Γ(x + s)<
?
x −1
2+
?
s +1
4
?1−s
(5)
for 0 < s < 1 and x ≥ 1. It is clear that the inequality (5) can be rearranged as
s
2<
Γ(x + s)
?Γ(x + 1)
?1/(1−s)
− x <
?
s +1
4−1
2. (6)
This suggests us to introduce a function
zs,t(x) =





?Γ(x + t)
eψ(x+s)− x,
Γ(x + s)
?1/(t−s)
− x,s ?= t
s = t
(7)
on x ∈ (−α,∞) for real numbers s and t and α = min{s,t}.
In [13, 22, 34, 35, 36, 39, 51, 52], the monotonic and convex properties of zs,t(x)
were established by using Laplace transform and other complicated techniques.
Their basic calculation is as follows:
z′
s,t(x) = [zs,t(x) + x]ψ(x + t) − ψ(x + s)
t − s
?2
− 1, (8)
z′′
s,t(x) = [zs,t(x) + x]
??ψ(x + t) − ψ(x + s)
[ψ(x + t) − ψ(x + s)]2+ (t − s)[ψ′(x + t) − ψ′(x + s)]
t − s
+ψ′(x + t) − ψ′(x + s)
t − s
?
(9)
=zs,t(x) + x
(t − s)2
This further suggests us to consider the following two functions:
??
. (10)
∆s,t(x) =





?ψ(x + t) − ψ(x + s)
[ψ′(x + s)]2+ ψ′′(x + s),
t − s
?2
+ψ′(x + t) − ψ′(x + s)
t − s
,s ?= t
s = t
(11)
and
Θs,t(x) = [ψ(x + t) − ψ(x + s)]2+ (t − s)[ψ′(x + t) − ψ′(x + s)]
on x ∈ (−α,∞) for real numbers s and t and α = min{s,t}.
In [3, p. 208], [8, Lemma 1.1] and [9, Lemma 1.1], the inequality
(12)
∆0,0(x) = [ψ′(x)]2+ ψ′′(x) > 0(13)

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A CLASS OF COMPLETELY MONOTONIC FUNCTIONS AND APPLICATIONS3
on (0,∞) was verified. In [4, 8, 9], this inequality was applied to provide some
symmetrical upper and lower bounds for Γ(x) in terms of ψ(x) as follows:
exp?α?eψ(x)(ψ(x) − 1) + 1??≤
where x∗= 1.4616··· denotes the only positive zero of ψ(x), and α and β are real
constants.
The first aim of this paper is to present the completely monotonic property of
the functions ∆s,t(x) and Θs,t(x) on (−α,∞), which implies the positivity of the
function ∆0,0(x) defined by (13).
Γ(x)
Γ(x∗)≤ exp?β?eψ(x)(ψ(x) − 1) + 1??, (14)
Theorem 1. The functions ∆s,t(x) for |t − s| < 1 and −∆s,t(x) for |t − s| > 1 are
completely monotonic on x ∈ (−α,∞). So are the functions Θs,t(x) for |t − s| < 1
and −Θs,t(x) for |t − s| > 1 on x ∈ (−α,∞).
The second aim of this paper is, by making use of Theorem 1, to provide an
alternative proof for the monotonicity and convexity of the function zs,t(x), which
is quoted as follows.
Theorem 2 ([13, 22, 36, 39, 51, 52]). The function zs,t(x) in (−α,∞) is either
convex and decreasing for |t − s| < 1 or concave and increasing for |t − s| > 1.
It is well-known [55] that Wallis cosine or sine formula is
?π/2
0
sinnxdx =
?π/2
0
cosnxdx
=
√π Γ((n + 1)/2)
nΓ(n/2)
=







π
2·(n − 1)!!
(n − 1)!!
n!!
n!!
for n even,
for n odd,
(15)
where n!! denotes the double factorial. It has been estimated by many mathemati-
cians and a lot of inequalities were established in, for example, [12, 14, 15, 16, 17,
18, 19, 20, 25, 27, 43, 54, 58] and related references therein.
The third aim of this paper is, by utilizing Theorem 2, to prove two sharp
double inequalities relating to Wallis cosine or sine formula (15) and to bound the
probability integral or error function as follows.
Theorem 3. For n ∈ N,
1
?π(n + 4/π − 1)
2?n + 9π/16 − 1
√π
?1 + (9π/16 − 1)/n
≤(2n − 1)!!
(2n)!!
<
1
?π(n + 1/4),
2?n + 3/4
(16)
√π
≤
(2n)!!
(2n + 1)!!<
√π
(17)
and
≤
?√n
−√n
e−x2dx <
√π
?1 − 3/(4n). (18)
In particular, taking n → ∞ in (18) leads to
?∞
−∞
e−x2dx =√π .(19)
The constants
best possible.
4
π−1 and1
4in (16) and the constants9π
16−1 and3
4in (17) are the

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4 F. QI AND B.-N. GUO
Let
Ωn=
πn/2
Γ(1 + n/2)
(20)
be the volume of the unit ball on Rn. The fourth aim of this paper is, by employing
Theorem 2, to recover a double inequality for ratio of the volumes of the unit balls
in Rn−1and Rnrespectively as follows.
Theorem 4 ([2, Theorem 2]). For n ∈ N, the inequality
?
holds if and only if A ≤1
The final aim of this paper is, by using Theorem 1, to generalize the inequal-
ity (14) to a monotonic property as follows.
n + A
2π
<Ωn−1
Ωn
≤
?
n + B
2π
(21)
2and B ≥π
2− 1.
Theorem 5. For real numbers s and t, α = min{s,t} and c ∈ (−α,∞), let



on x ∈ (−α,∞). Then the function


on x ∈ (−α,∞) is decreasing for |s − t| < 1 and increasing for |s − t| > 1.
Remark 1. If taking c = x∗, then the case s = t in Theorem 5 becomes [4, Theo-
rem 4.3]: For 0 < a < b ≤ ∞ and x ∈ (a,b), the inequality (14) holds with the best
possible constant factors
gs,t(x) =




1
t − s
?x
?x
c
ln
?Γ(u + t)
Γ(u + s)
Γ(c + s)
Γ(c + t)
?
du,s ?= t
c
[ψ(u + s) − ψ(c + s)]du,s = t
(22)
fs,t(x) =





gs,t(x)
[g′
s,t(x) − 1]exp[g′
1
g′′
s,t(x)] + 1,x ?= c
s,t(c),x = c
(23)
α =
?
Q(b),
1,
b < ∞
b = ∞
andβ = Q(a), (24)
where
Q(x) =







lnΓ(x) − lnΓ(x∗)
eψ(x)[ψ(x) − 1] + 1,
1
ψ′(x∗),
x ?= x∗;
x = x∗.
(25)
2. Lemmas
In order to prove our theorems, the following lemmas are necessary.
Lemma 1. Let f(x) be defined in an infinite interval I. If
lim
x→∞f(x) = δ
and
f(x) − f(x + ε) > 0
for some given ε > 0, then f(x) > δ on I.
Proof. By induction, for any x ∈ I, we have
f(x) > f(x + ε) > f(x + 2ε) > ··· > f(x + kε) → δ
as k → ∞. The proof of Lemma 1 is complete.
?

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A CLASS OF COMPLETELY MONOTONIC FUNCTIONS AND APPLICATIONS5
Lemma 2 ([1]). For any positive integer n ∈ N and x > 0,
ψ(x) = lnx +
?∞
0
?1
u−
?∞
1
1 − e−u
un
1 − e−ue−xudu,
?
e−xudu,(26)
ψ(n)(x) = (−1)n+1
ψ(n−1)(x + 1) = ψ(n−1)(x) +(−1)n−1(n − 1)!
0
(27)
xn
, (28)
lnx −1
x< ψ(x) < lnx −
1
2x.
(29)
As x → ∞,
ψ′(x) ∼1
x+
1
2x2+ ··· . (30)
Lemma 3 ([37, 38]). For s > r > 0,
exp[(s − r)ψ(s)] >Γ(s)
Γ(r)> exp[(s − r)ψ(r)]. (31)
Lemma 4 ([57]). A product of finite completely monotonic functions is also com-
pletely monotonic.
3. Proofs of theorems
Now we are in a position to prove our theorems.
Proof of Theorem 1. Direct computation and utilization of (28) gives
Θs,t(x) − Θs,t(x + 1) =?[ψ(x + t) + ψ(x + t + 1)] − [ψ(x + s) + ψ(x + s + 1)]?
×?[ψ(x + t) − ψ(x + t + 1)] − [ψ(x + s) − ψ(x + s + 1)]?
+ (t − s)?[ψ′(x + t) − ψ′(x + t + 1)] − [ψ′(x + s) − ψ′(x + s + 1)]?
=
?[ψ(x + t + 1) + ψ(x + t)] − [ψ(x + s + 1) + ψ(x + s)]
2x + s + t
(x + s)(x + t)(x + s)(x + t)
(t − s)2
(x + s)(x + t)
t − s
−
?
(t − s)2
? Λs,t(x)
(32)
and
Λs,t(x) − Λs,t(x + 1) =
1
t − s
−2x2+ 2(s + t + 1)x + s2+ t2+ s + t
(x + s)(x + s + 1)(x + t)(x + t + 1)
1 − (s − t)2
(x + s)(x + s + 1)(x + t)(x + t + 1).
?
1
x + s+
1
x + s + 1−
1
x + t−
1
x + t + 1
?
=
Since limx→∞Λ(i)
function
s,t(x) = 0 for any nonnegative integer i by (26) and (27) and the
Λs,t(x) − Λs,t(x + 1)
1 − (s − t)2
is completely monotonic by Lemma 4, that is,
(−1)i[Λs,t(x) − Λs,t(x + 1)](i)
1 − (s − t)2
=(−1)iΛ(i)
s,t(x) − (−1)iΛ(i)
1 − (s − t)2
s,t(x + 1)
≥ 0,