Some two-sided inequalities for multiple Gamma functions and related ...
Author's personal copy Applied Mathematics and Computation 219 (2013) 10343–10354
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some two-sided inequalities for multiple Gamma functions and related results Junesang Choi a, H.M. Srivastava b,⇑ a b
Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4 Canada
a r t i c l e
i n f o
a b s t r a c t There is an abundant literature on inequalities for the (Euler’s) Gamma function C and its various related functions. Yet, only very recently, several authors began to study inequalities for the (Barnes’) double Gamma function C2 . Here, in this paper, we aim at presenting several two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ. In our investigation of these two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ, we employ and extend a method based upon Taylor’s formula and express log Cn ð1 þ xÞ as series involving the Zeta functions. We also give a more convenient explicit form of the multiple Gamma functions Cn ðn 2 NÞ; N being the set of positive integers. The main two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions Cn when n=6. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: Gamma, Psi (or Digamma) and Polygamma functions Double and multiple Gamma functions Riemann and Hurwitz (or generalized) Zeta functions Bohr–Mollerup theorem Harmonic numbers and the Stirling numbers of the first kind Euler–Mascheroni and Glaisher–Kinkelin constants Determinants of the Laplacians and Weierstrass canonical product forms Series involving the Zeta functions
1. Introduction, definitions and preliminaries The multiple Gamma functions Cn were defined and studied systematically by Barnes [9–12] and by others (cf., e.g., [4,38–40]) in about 1900 (see also [37, p. 649, Entry 6.441(4); p. 887, Entry 8.333] and [63, p. 264]). About two decades ago, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere Sn (cf. [20,21,25,26,43,59,61]) and have since been investigated in various other ways (cf. [57, p. 24, Section 1.3] and [58, p. 38, Section 1.4]); see also [2,3,15–17,22,23,27,35,41,44–49,52,54]. There is a remarkably abundant literature on inequalities for the classical Gamma function C and its such related functions as, for example, the Psi (or Digamma) function defined by
wðzÞ :¼
d C0 ðzÞ flog CðzÞg ¼ dz CðzÞ
or
log CðzÞ ¼
Z
z
wðsÞds;
1
(see, e.g., [5,6,50,51]; see also the references cited in these earlier works). Among several equivalent useful expressions for the Gamma function, its Weierstrass canonical product form is recalled here as follows: 1 z ecz Y z 1 CðzÞ ¼ 1þ exp z 2 C n Z0 ; k k z k¼1
⇑ Corresponding author. E-mail addresses: [email protected] (J. Choi), [email protected] (H.M. Srivastava). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.006
ð1:1Þ
Author's personal copy 10344
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
where C is the set of complex numbers, Z 0 :¼ f0; 1; 2; g and c denotes the Euler–Mascheroni constant defined by (see also a recent work [29])
c :¼ n!1 lim
n X 1 k¼1
k
! ffi 0:57721 56649 01532 86060 65120 90082 40243 1042 . . . :
log n
ð1:2Þ
On the other hand, there are only a few recent papers on inequalities for the double Gamma function C2 :¼ 1=G (see, for example, the recent works by Batir [13], Batir and Cancan [14], Chen [18], and Chen and Srivastava [19]). Very recently, Koumandos and Pedersen [42] and Choi and Srivastava [31] presented asymptotic formulas for Barnes’ triple Gamma function. Barnes [9] gave several explicit Weierstrass canonical product forms of the double Gamma function C2 :¼ 1=G, one of which is recalled here as follows:
Y 1 1 1 z k z2 ; fC2 ðz þ 1Þg1 ¼ Gðz þ 1Þ ¼ ð2pÞz=2 exp z ðc þ 1Þz2 1þ exp z þ 2 2 k 2k k¼1
ð1:3Þ
where c denotes the Euler–Mascheroni constant given by Eq. (1.2). Analogous to the familiar relations:
Cð1Þ ¼ 1 and Cðz þ 1Þ ¼ z CðzÞ;
ð1:4Þ
the double Gamma function C2 :¼ 1=G satisfies the following fundamental relations:
Gð1Þ ¼ 1 and Gðz þ 1Þ ¼ CðzÞ GðzÞ:
ð1:5Þ
Like the Euler–Mascheroni constant c in Eq. (1.2), there is a set of constants which are naturally involved in the theory of multiple Gamma functions (see [1,15,36] and [57, p. 128]). Some of these constants are introduced here for our latter use. First of all, A denotes the Glaisher–Kinkelin constant defined by (see [9])
"
# 2 n n 1 n2 ffi 1:282427130 . . . : log n þ log A ¼ lim k log k þ þ n!1 2 2 12 4 k¼1 n X
ð1:6Þ
The constants B and C are analogous to the Glaisher–Kinkelin constant A and are defined by (see [26])
" # 3 n X n n2 n n3 7 2 log n þ log B ¼ lim k log k þ þ n!1 3 2 6 9 12 k¼1
ð1:7Þ
and
"
# 4 n X n n3 n2 1 n4 n2 3 ; log n þ log C ¼ lim k log k þ þ n!1 4 2 4 120 16 12 k¼1
ð1:8Þ
respectively. The approximate numerical values of the constant B and C are given by
B ffi 1:03091675
and C ffi 0:97955746 . . . :
The constants A; B and C are also known to be expressible as follows:
log A ¼
1 f0 ð1Þ; 12
log B ¼ f0 ð2Þ;
and
log C ¼
11 f0 ð3Þ; 720
ð1:9Þ
in terms of special values of the derivative of the Riemann zeta function fðsÞ defined by
fðsÞ :¼
8 1 1 X X > > 1 1 1 > s ¼ 12s > n ð2n1Þs < n¼1
RðsÞ > 1 ;
n¼1
1 X > > ð1Þn1 1 > > 1s : 12 ns
RðsÞ > 0; s – 1 :
ð1:10Þ
n¼1
The Riemann Zeta function fðsÞ is a special case of the Hurwitz (or generalized) Zeta function fðs; aÞ defined by
fðs; aÞ :¼
1 X k¼0
1 ðk þ aÞs
RðsÞ > 1; a 2 C n Z0 ;
ð1:11Þ
each of which can be continued meromorphically to the whole complex s-plane except for a simple pole at s ¼ 1 with its residue 1 (see, for details, [57, pp. 88–103]). Remark 1. Adamchik [1] (see also [15] and [57, p. 128]) presented a set of mathematical constants which include the abovedefined A; B and C as special cases. The Polygamma functions wðnÞ ðzÞ ðn 2 NÞ are defined by
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354 nþ1
wðnÞ ðzÞ :¼
d
nþ1
dz
flog CðzÞg ¼
n d n 2 N0 ; z 2 C n Z0 ; n fwðzÞg dz
10345
ð1:12Þ
where N is the set of positive integers and N0 :¼ N [ f0g. In terms of the Hurwitz (or generalized) Zeta function fðs; aÞ, we can write
wðnÞ ðzÞ ¼ ð1Þnþ1 n!
1 X
1
k¼0
ðk þ zÞnþ1
¼ ð1Þnþ1 n! fðn þ 1; zÞ ðn 2 N; z 2 C n Z0 Þ;
ð1:13Þ
which has turned out to be useful in certain applications (see, e.g., [28]). Here, in this paper, we aim at presenting some two-sided inequalities for the multiple Gamma functions Cn for (especially) n ¼ 2; 3; 4; 5. We employ and extend the method initiated by Batir and Cancan [14] and express log Cn ð1 þ xÞ as series involving the Zeta functions (see, e.g., [7,21,24–26,33,53,55–57]). We also derive a more convenient and potentially useful explicit form of the multiple Gamma functions Cn ðn 2 NÞ. The main two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions Cn when n P q6. 2. Multiple gamma functions There are two known ways to define the n-ple Gamma functions Cn . First of all, Barnes [12] (see also Vardi [59]) defined Cn by using the n-ple Hurwitz Zeta functions (see, e.g., [22], [57, Chapter 2]). Secondly, a recurrence relation of the Weierstrass canonical product forms of the n-ple Gamma functions Cn was given by Vignéras [60] who used the theorem of Dufresnoy and Pisot [34] which provides the existence, uniqueness, and expansion of the series of Weierstrass satisfying a certain functional equation. By making use of the aforementioned Dufresnoy–Pisot theorem and starting with
f1 ðxÞ ¼ c x þ
1 h X x
x i log 1 þ ; n n
n¼1
ð2:1Þ
Vignéras [60] obtained a recurrence relation of Cn ðn 2 NÞ which is stated here as Theorem 1 below. Theorem 1. The n-ple Gamma functions Cn are defined by
Cn ðzÞ ¼ ½Gn ðzÞð1Þ
n1
ðn 2 NÞ;
ð2:2Þ
where
Gn ðz þ 1Þ ¼ exp ½fn ðzÞ
ð2:3Þ
and the functions fn ðzÞ are given by
fn ðzÞ ¼ z An ð1Þ þ
n1 i X pk ðzÞ h ðkÞ fn1 ð0Þ AnðkÞ ð1Þ þ An ðzÞ; k! k¼1
ð2:4Þ
with
An ðzÞ ¼
X m2Nn1 N 0
" n n1 # 1 z 1 z z z n1 n ; þ ð1Þ log 1 þ þ þ ð1Þ n LðmÞ n 1 LðmÞ LðmÞ LðmÞ
ð2:5Þ
where
LðmÞ ¼ m1 þ m2 þ m3 þ þ mn ; if
m ¼ ðm1 ; m2 ; m3 ; ; mn Þ 2 Nn1 N 0 and the polynomials pn ðzÞ given by
8 n n n n < 1 þ 2 þ 3 þ þ ðN 1Þ pn ðzÞ :¼ Bnþ1 ðzÞ Bnþ1 : nþ1
satisfy the following relations:
p0n ðzÞ ¼
B0nþ1 ðzÞ ¼ Bn ðzÞ and pn ð0Þ ¼ 0; nþ1
ðz ¼ N; N 2 N n f1gÞ; ðz 2 CÞ;
ð2:6Þ
Author's personal copy 10346
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
Bn ðzÞ being the Bernoulli polynomial of degree n in z. By analogy with the Bohr–Mollerup theorem (see [8, p. 14]; see also [57, p. 13]), which guarantees the uniqueness of the Gamma function C, one can give, for the double Gamma function and (more generally) for the multiple Gamma functions of order n ðn 2 NÞ, a definition of Artin [8] by means of the following theorem (see Vignéras [60, p. 239]). Theorem 2. For all n 2 N, there exists a unique meromorphic function Gn ðzÞ satisfying each of the following properties: (a) Gn ðz þ 1Þ ¼ Gn1 ðzÞ Gn ðzÞ ðz 2 CÞ; (b) Gn ð1Þ ¼ 1; (c) For x=1; Gn ðxÞ are infinitely differentiable and nþ1
d
nþ1
dx
flog Gn ðxÞg=0;
(d) G0 ðxÞ ¼ x. It is not difficult to verify (see, e.g., [57, pp. 40–41]) that fCn ðzÞg1 is an entire function with zeros at z ¼ k ðk 2 N0 Þ with multiplicities
nþk1
ðn 2 N; k 2 N0 Þ:
n1
ð2:7Þ
In our earlier investigations, we gave explicit forms of the multiple Gamma functions Cn ðn ¼ 3; 4; 5Þ (see, e.g., [21,33]). Now, by observing Eq. (2.7), we can present the following explicit form of the multiple Gamma functions Cn ðn 2 NÞ for a potential and easier future use. Theorem 3. The n-ple Gamma functions Cn in Theorem 1 can be written in a more explicit form as follows:
8 9 nþk2 > " !#> > > 1 > k j n1 k > j¼1 k¼1 > : ;
ð2:8Þ
where Q n ðzÞ is a polynomial in z of degree n given by
"
n1
Q n ðzÞ :¼ ð1Þ
# n1 X pk ðzÞ ðkÞ ðkÞ z An ð1Þ þ fn1 ð0Þ An ð1Þ ; k! k¼1
fn ðzÞ :¼ z An ð1Þ þ
An ðzÞ :¼
1 X k¼1
ð1Þ
n1 i X pk ðzÞ h ðkÞ fn1 ð0Þ AnðkÞ ð1Þ þ An ðzÞ; k! k¼1
n1
nþk2 n1
# " n z X ð1Þj1 zj log 1 þ þ j k j k j¼1
ð2:9Þ
ð2:10Þ
ð2:11Þ
and
pn ðzÞ ¼
nþ1 nþ1 1 X Bnþ1k zk n þ 1 k¼1 k
ðn 2 NÞ:
ð2:12Þ
Remark 2. In order to get explicit forms of the multiple Gamma functions Cn ð1 þ zÞ in Theorem 3, it is indispensable to compute An ð1Þ explicitly. In fact, by using the Taylor–Maclaurin expansion of logð1 þ tÞ in Eq. (2.11) and certain series involving Zeta functions, Choi et al. [21] found that
" j j1 n1 X X X j 0 1 fð‘Þ j‘ zjþ1 f ðk; 1 þ zÞ zjk þ ð1Þjþ1 f0 ðjÞ An ðzÞ ¼ sðn 1; jÞ ð1Þk ð1Þ‘ z þ Hj þ c ðn 1Þ! j¼0 j‘ jþ1 k ‘¼0 k¼0 # nj X fðkÞ kþj ; ð2:13Þ ð1Þk z k þj k¼2 where sðn; kÞ denotes the Stirling numbers of the first kind (see [57, pp. 56–57]) and Hn denotes the harmonic numbers given by
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
Hn :¼
n X 1 k¼1
k
10347
ðn 2 NÞ:
Now, by applying Eq. (2.13) in Theorem 3, we can give explicit forms of the multiple Gamma functions Cn ðn 2 NÞ whose cases ðn ¼ 3; 4; 5Þ are recalled here as the following corollary (see [21]). Corollary 1. Each of the following expressions holds true:
8 9 kþ1 > > > >
1 > k k 2k 2 3k > k¼1 > : ;
ð2:14Þ
where
1 1 c 1 p2 c þ logð2pÞ þ ; c3 ¼ ; 8 4 4 36 6 4 8 9 kþ2 > > > >
1 < = Y kþ2 z z z2 z3 z4 2 3 4 3 C4 ð1 þ zÞ ¼ exp d1 z þ d2 z þ d3 z þ d4 z 1þ 2þ 3 4 ; exp > > k k 2k 3 3k 4k > k¼1 > : ;
c1 ¼
3 1 logð2pÞ log A; 8 4
c2 ¼
ð2:15Þ
where
d1 ¼
7 1 1 log A log B logð2pÞ; 24 2 6
2 c 1 p2 d3 ¼ ; logð2pÞ 9 6 12 54
d2 ¼
1 c 1 1 þ þ logð2pÞ þ log A; 144 6 4 2
11 c p2 fð3Þ þ þ þ ; 144 24 48 12 8 9 kþ3 > > > >
1 < = 2 3 4 5 Y kþ3 z z z z z z 2 3 4 5 4 C5 ð1 þ zÞ ¼ exp e1 z þ e2 z þ e3 z þ e4 z þ e5 z 1þ 2þ 3 4þ 5 ; exp > > k k 2k 3k 4k 5k 4 > k¼1 > : ; d4 ¼
ð2:16Þ where
e1 ¼
409 1 11 1 3 fð3Þ 1 1 log A log C logð2pÞ þ fð4Þ fð5Þ; 1728 8 12 6 16 p2 20 20
e2 ¼
1 c 11 3 fð3Þ þ þ ; logð2pÞ þ log A þ 16 8 48 4 16 p2
e3 ¼
149 11 1 1 1 c logð2pÞ log A fð2Þ; 864 72 8 6 12
e4 ¼
7 1 1 11 1 þ c þ logð2pÞ þ fð2Þ þ fð3Þ; 64 16 48 96 16
e5 ¼
5 1 1 11 1 c fð2Þ fð3Þ fð4Þ: 288 120 20 120 20
3. Use of the method based upon Taylor’s formula By applying Taylor’s formula (see, e.g., [62, p. 209, Theorem 7.44]) to the function hðtÞ given by
hðtÞ ¼ t log t
ðt 2 ½k; k þ xÞÞ
up to its third derivative, Batir and Cancan [14] found that
x x x2 x3 log 1 þ ¼ 2þ k k 2k 3ðk þ lðkÞÞ3 where
ð0 < lðkÞ < xÞ;
lðkÞ can be easily expressed as follows:
ð3:1Þ
Author's personal copy 10348
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
lðkÞ ¼
13 3 x 3 3 log 1 þ þ k: x3 k k x2 2k2 x
Batir and Cancan [14] proved that
ð3:2Þ
lðkÞ is strictly increasing for all k=1 and x > 0 and that
x lð1Þ ¼ lim lðkÞ ¼ : k!1 4 By using these observations, Batir and Cancan [14] derived a two-sided inequality for the double Gamma function G ¼ 1=C2 . More generally, by applying Taylor’s formula to the same function as above to get the remainder on ½k; k þ xÞ of the nth derivative ðn 2 NÞ, we obtain n1 x X ð1Þj1 xj ð1Þn1 xn log 1 þ ;þ ¼ n k k j n k þ ln ðk; xÞ j¼1
ð3:3Þ
k=1; x > 0; n 2 N; 0 < ln ðk; xÞ < x ;
ln ðk; xÞ can be written in the following explicit form:
which shows that
1
ln ðk; xÞ ¼ fg n ðk; xÞgn k ðk=1; x > 0; n 2 NÞ;
ð3:4Þ
where, for convenience,
" # n1 ð1Þn1 n x X ð1Þj1 xj : g n ðk; xÞ :¼ log 1 þ xn k k j j¼1 We now provide some properties of
ð3:5Þ
ln ðk; xÞ as asserted by the following lemmas.
Lemma 1. The following formula holds true:
lim ln ðk; xÞ ¼
k!1
x nþ1
ðx > 0; n 2 NÞ:
ð3:6Þ
Proof. In view of the Taylor–Maclaurin series expansion of logð1 þ tÞ, we find, for any bounded real number x, that
n1 1 x X ð1Þj1 xj X ð1Þj1 xj ð1Þn1 xn ð1Þn xnþ1 1 ðk ! 1Þ; log 1 þ ¼ ¼ þ O nþ2 n þ k k k j j n n þ 1 knþ1 k k j¼1 j¼n so that
ln ðk; xÞ ¼
1n
1n 1 nx 1 1 nx 1 1 þ O k ¼ k 1 þ O k ðk ! 1Þ: n nþ1 nþ2 2 nþ1 k nþ1 k k k k
Using the generalized binomial theorem, we get
ln ðk; xÞ ¼ k
1 X
1n j
j¼0
!
j nx 1 1 þO 2 k; nþ1 k k
where the generalized binomial coefficient
a j
¼
8 0.
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
Lemma 3. The function
10349
ln ðk; xÞ is strictly increasing on k 2 ½1; 1Þ and x > 0 for all n 2 N.
Proof. In view of Eq. (3.5), by considering the sum formula of a finite geometric series, we get
@ n fg ðk; xÞg ¼ n @k n k ðk þ xÞ
ðn 2 NÞ;
which readily yields
@ 1 nþ1 fl ðk; xÞg ¼ n fg n ðk; xÞg n 1: @k n k ðk þ xÞ
ln ðk; xÞ is strictly increasing on k 2 ½1; 1Þ and x > 0 if and only if
It is thus observed that the function
@ k 2 ½1; 1Þ; x > 0 ; fln ðk; xÞg > 0 @k that is, if and only if
( ð1Þ
n1
#)nþ1 n " n1 k x X ð1Þj1 xj 1 n log 1 þ 0 : Here, by setting kx ¼ t in Eq. (3.7), it is seen that the function ln ðk; xÞ is strictly increasing on k 2 ½1; 1Þ and x > 0 if and only if nþ1
Hn ðtÞ :¼ ½hn ðtÞ
t nðnþ1Þ < 0 ðt > 0; n 2 NÞ; nnþ1 ð1 þ tÞn
ð3:8Þ
where, for convenience,
hn ðtÞ :¼ ð1Þ
n1
logð1 þ tÞ
n1 X ð1Þj1 j¼1
j
! t
j
ðt > 0Þ:
We note that
tn1 1þt
0
hn ðtÞ ¼
ðt > 0; n 2 NÞ:
It is observed that, for the inequality in Eq. (3.8), it is equivalent to prove that
Gn ðtÞ :¼ hn ðtÞ
tn n
n ð1 þ tÞnþ1
< 0 ðt > 0; n 2 NÞ:
ð3:9Þ
We now want to prove the following inequality:
G0n ðtÞ ¼
t n1 tn t n1 þ ðt > 0; n 2 NÞ; n n < 0 1þ 1 þ t ðn þ 1Þ ð1 þ tÞ nþ1 ð1 þ tÞnþ1
if and only if
1þ
t
1
n
ðn þ 1Þ ð1 þ tÞnþ1
ð1 þ tÞnþ1 < 0 ðt > 0; n 2 NÞ;
that is, if and only if n
ð1 þ tÞnþ1 þ
t 0; n 2 NÞ;
that is, if and only if n
ð1 þ tÞnþ1 < 1 þ
n t nþ1
ðt > 0; n 2 NÞ;
the last inequality of which holds true by means of the first part of the Bernoulli inequality in Lemma 2. Thus Gn ðtÞ is strictly decreasing on t 2 ð0; 1Þ for all n 2 N, and so we have
Gn ðtÞ < Gn ð0Þ ¼ 0 ðt > 0; n 2 NÞ: This completes the proof of the inequality in Eq. (3.9).
h
Author's personal copy 10350
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
From Lemmas 1 and 3, it is easy to deduce Lemma 4 below. Lemma 4. The following two-sided inequality holds true:
ln ð1; xÞ 5 ln ðk; xÞ
0; n 2 N :
ð3:10Þ
For our convenience and later use, we define the following two functions:
an ðxÞ :¼ 1 þ
x nþ1
and bn ðxÞ :¼ 1 þ ln ð1; xÞ ðx > 0; n 2 NÞ:
ð3:11Þ
We are now ready to present our proposed two-sided inequalities for the multiple Gamma functions Cn ð1 þ xÞ ðn ¼ 2; ; 5; x > 0Þ. Theorem 4. Each of the following two-sided inequalities holds true:
3 3 x x x x ð2pÞ2 exp ½p2 ðxÞ exp qb3 ðxÞ 5 C2 ð1 þ xÞ 5 ð2pÞ2 exp ½p2 ðxÞ exp qa3 ðxÞ ; 3 3
ð3:12Þ
where
x 1 þ ðc þ 1Þx2 ; 2 2 qa3 ðxÞ :¼ f 2; a3 ðxÞ þ ½1 a3 ðxÞ f 3; a3 ðxÞ ; p2 ðxÞ :¼
qb3 ðxÞ :¼ f 2; b3 ðxÞ þ ½1 b3 ðxÞ f 3; b3 ðxÞ ; 4 4 x x exp ½p3 ðxÞ exp q ðxÞ 5 C3 ð1 þ xÞ 5 exp ½p3 ðxÞ exp q ðxÞ ; 8 a4 8 b4
ð3:13Þ
where
p3 ðxÞ :¼ c1 x þ c2 x2 þ c3 x3 ;
qa4 ðxÞ :¼ f 2; a4 ðxÞ þ ½1 2a4 ðxÞ f 3; a4 ðxÞ þ a24 ðxÞ a4 ðxÞ f 4; a4 ðxÞ ;
qb4 ðxÞ :¼ f 2; b4 ðxÞ þ ½1 2b4 ðxÞ f 3; b4 ðxÞ þ b24 ðxÞ b4 ðxÞ f 4; b4 ðxÞ ; x5 x5 exp ½p4 ðxÞ exp qb5 ðxÞ 5 C4 ð1 þ xÞ 5 exp ½p4 ðxÞ exp qa5 ðxÞ ; 30 30
ð3:14Þ
where
p4 ðxÞ :¼ d1 x þ d2 x2 þ d3 x3 þ d4 x4 ;
qa5 ðxÞ :¼ f 2; a5 ðxÞ þ ½6 3a5 ðxÞ f 3; a5 ðxÞ þ 11 12 a5 ðxÞ þ 3 a25 ðxÞ f 4; a5 ðxÞ þ ½1 a5 ðxÞ½2 a5 ðxÞ½3 a5 ðxÞ f 5; a5 ðxÞ ; qb5 ðxÞ :
¼ f 2; b5 ðxÞ þ ½6 3b5 ðxÞ f 3; b5 ðxÞ þ 11 12 b5 ðxÞ þ 3 b25 ðxÞ f 4; b5 ðxÞ þ ½1 b5 ðxÞ½2 b5 ðxÞ½3 b5 ðxÞ f 5; b5 ðxÞ ;
exp ½p5 ðxÞ exp
6 x6 x qa6 ðxÞ 5 C5 ð1 þ xÞ 5 exp ½p5 ðxÞ exp qb6 ðxÞ ; 144 144
where
p5 ðxÞ :¼ e1 x þ e2 x2 þ e3 x3 þ e4 x4 þ e5 x5 ;
ð3:15Þ
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
10351
qa6 ðxÞ :
¼ f 2; a6 ðxÞ þ ½10 4a6 ðxÞ f 3; a6 ðxÞ þ 35 30 a6 ðxÞ þ 6 a26 ðxÞ f 4; a6 ðxÞ
þ 50 70 a6 ðxÞ þ 30 a26 ðxÞ 4 a36 ðxÞ f 5; a6 ðxÞ þ ½1 a6 ðxÞ½2 a6 ðxÞ½3 a6 ðxÞ½4 a6 ðxÞ f 5; a6 ðxÞ ;
qb6 ðxÞ :
¼ f 2; b6 ðxÞ þ ½10 4b6 ðxÞ f 3; b6 ðxÞ þ 35 30 b6 ðxÞ þ 6 b26 ðxÞ f 4; b6 ðxÞ
þ 50 70 b6 ðxÞ þ 30 b26 ðxÞ 4 b36 ðxÞ f 5; b6 ðxÞ þ ½1 b6 ðxÞ½2 b6 ðxÞ½3 b6 ðxÞ½4 b6 ðxÞ f 5; b6 ðxÞ :
Proof. Upon taking logarithms on each of the explicit Weierstrass canonical product forms Eqs. (1.3), (2.14), (2.15) and (2.16) of the multiple Gamma functions and applying the result in Lemma 4, if we make a little simplification, we are led to the inequalities asserted by Theorem 4. h Remark 3. In view of the relation Eq. (1.13), the Hurwitz (or generalized) Zeta function fðs; aÞ in Theorem 4 can be replaced by the Polygamma functions wðnÞ ðzÞ as it was done in [14]. 4. Use of series involving the zeta functions Ever since the Goldbach theorem of 1729 (see [53]; see also [57, p. 142]), closed-form evaluations of series involving the Zeta functions have not only attracted many important investigations (see, e.g., [7,21,24–26,33,53,55,57]), but have also found their applications in various ways (see, e.g., [21,25,30]). Our very recent paper [32] deals extensively with the familiar family of the Goldbach–Euler series; in fact, it was motivated essentially by the various developments emerging from the aforementioned Goldbach theorem of 1729. In light of Eq. (1.10), if we take logarithms of both sides in Eqs. (1.3), (2.14), (2.15) and (2.16) and apply the TaylorMaclaurin expansion of logð1 þ tÞ to each of the resulting identities, we find the series representations given by Theorem 5 below. Theorem 5. Each of the following series representations holds true:
log C2 ð1 þ zÞ ¼ c2;1 z þ c2;2 z2 þ
1 X ð1Þkþ1 k¼2
kþ1
log C3 ð1 þ zÞ ¼ c3;1 z þ c3;2 z2 þ c3;3 z3 þ
fðkÞ zkþ1 ;
1 1 1X ð1Þkþ1 1X ð1Þk fðkÞ zkþ1 þ fðkÞ zkþ2 ; 2 k¼2 k þ 1 2 k¼2 k þ 2
log C4 ð1 þ zÞ ¼ c4;1 z þ c4;2 z2 þ c4;3 z3 þ c4;4 z4 þ þ
1 1X ð1Þkþ1 fðnÞzkþ3 ; 6 k¼2 k þ 3
ð4:3Þ 1 1 1X ð1Þkþ1 11 X ð1Þk fðkÞzkþ1 þ fðkÞzkþ2 4 k¼2 k þ 1 24 k¼2 k þ 2
1 1 1X ð1Þkþ1 1 X ð1Þk fðkÞzkþ3 þ fðkÞzkþ4 ; 4 k¼2 k þ 3 24 k¼2 k þ 4
where
c2;1 ¼
1 ½1 logð2pÞ; 2
c3;2 ¼
1 1 c þ logð2pÞ þ ; 8 4 4
c4;1 ¼
7 1 1 logð2pÞ log A log B; 24 6 2
c2;2 ¼
ð4:2Þ
1 1 1X ð1Þkþ1 1X ð1Þk fðkÞ zkþ1 þ fðkÞ zkþ2 3 k¼2 k þ 1 2 k¼2 k þ 2
log C5 ð1 þ zÞ ¼ c5;1 z þ c5;2 z2 þ c5;3 z3 þ c5;4 z4 þ c5;5 z5 þ þ
ð4:1Þ
1 ðc þ 1Þ; 2
c3;1 ¼
3 1 logð2pÞ log A; 8 4
1 c c3;3 ¼ ; 4 6 c4;2 ¼
1 c 1 1 þ þ logð2pÞ þ log A; 144 6 4 2
ð4:4Þ
Author's personal copy 10352
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
2 1 c c4;3 ¼ logð2pÞ ; 9 12 6
c4;4 ¼
c5;1 ¼ e1 ;
149 11 1 1 c logð2pÞ log A; 864 72 8 6
c5;4 ¼
c5;2 ¼ e2 ;
c5;3 ¼
7 c 1 þ þ logð2pÞ; 64 16 48
11 c þ ; 144 24
5 c þ : 288 120
c5;5 ¼
We next recall here the well-known property of an alternating series of real numbers. Lemma 5. Let fak g1 k¼1 be a decreasing sequence of positive real numbers with ak # 0 as k ! 1. Then the following inequalities hold true: 2n X
ð1Þkþ1 ak
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some two-sided inequalities for multiple Gamma functions and related results Junesang Choi a, H.M. Srivastava b,⇑ a b
Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4 Canada
a r t i c l e
i n f o
a b s t r a c t There is an abundant literature on inequalities for the (Euler’s) Gamma function C and its various related functions. Yet, only very recently, several authors began to study inequalities for the (Barnes’) double Gamma function C2 . Here, in this paper, we aim at presenting several two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ. In our investigation of these two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ, we employ and extend a method based upon Taylor’s formula and express log Cn ð1 þ xÞ as series involving the Zeta functions. We also give a more convenient explicit form of the multiple Gamma functions Cn ðn 2 NÞ; N being the set of positive integers. The main two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions Cn when n=6. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: Gamma, Psi (or Digamma) and Polygamma functions Double and multiple Gamma functions Riemann and Hurwitz (or generalized) Zeta functions Bohr–Mollerup theorem Harmonic numbers and the Stirling numbers of the first kind Euler–Mascheroni and Glaisher–Kinkelin constants Determinants of the Laplacians and Weierstrass canonical product forms Series involving the Zeta functions
1. Introduction, definitions and preliminaries The multiple Gamma functions Cn were defined and studied systematically by Barnes [9–12] and by others (cf., e.g., [4,38–40]) in about 1900 (see also [37, p. 649, Entry 6.441(4); p. 887, Entry 8.333] and [63, p. 264]). About two decades ago, these functions were revived in the study of the determinants of the Laplacians on the n-dimensional unit sphere Sn (cf. [20,21,25,26,43,59,61]) and have since been investigated in various other ways (cf. [57, p. 24, Section 1.3] and [58, p. 38, Section 1.4]); see also [2,3,15–17,22,23,27,35,41,44–49,52,54]. There is a remarkably abundant literature on inequalities for the classical Gamma function C and its such related functions as, for example, the Psi (or Digamma) function defined by
wðzÞ :¼
d C0 ðzÞ flog CðzÞg ¼ dz CðzÞ
or
log CðzÞ ¼
Z
z
wðsÞds;
1
(see, e.g., [5,6,50,51]; see also the references cited in these earlier works). Among several equivalent useful expressions for the Gamma function, its Weierstrass canonical product form is recalled here as follows: 1 z ecz Y z 1 CðzÞ ¼ 1þ exp z 2 C n Z0 ; k k z k¼1
⇑ Corresponding author. E-mail addresses: [email protected] (J. Choi), [email protected] (H.M. Srivastava). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.006
ð1:1Þ
Author's personal copy 10344
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
where C is the set of complex numbers, Z 0 :¼ f0; 1; 2; g and c denotes the Euler–Mascheroni constant defined by (see also a recent work [29])
c :¼ n!1 lim
n X 1 k¼1
k
! ffi 0:57721 56649 01532 86060 65120 90082 40243 1042 . . . :
log n
ð1:2Þ
On the other hand, there are only a few recent papers on inequalities for the double Gamma function C2 :¼ 1=G (see, for example, the recent works by Batir [13], Batir and Cancan [14], Chen [18], and Chen and Srivastava [19]). Very recently, Koumandos and Pedersen [42] and Choi and Srivastava [31] presented asymptotic formulas for Barnes’ triple Gamma function. Barnes [9] gave several explicit Weierstrass canonical product forms of the double Gamma function C2 :¼ 1=G, one of which is recalled here as follows:
Y 1 1 1 z k z2 ; fC2 ðz þ 1Þg1 ¼ Gðz þ 1Þ ¼ ð2pÞz=2 exp z ðc þ 1Þz2 1þ exp z þ 2 2 k 2k k¼1
ð1:3Þ
where c denotes the Euler–Mascheroni constant given by Eq. (1.2). Analogous to the familiar relations:
Cð1Þ ¼ 1 and Cðz þ 1Þ ¼ z CðzÞ;
ð1:4Þ
the double Gamma function C2 :¼ 1=G satisfies the following fundamental relations:
Gð1Þ ¼ 1 and Gðz þ 1Þ ¼ CðzÞ GðzÞ:
ð1:5Þ
Like the Euler–Mascheroni constant c in Eq. (1.2), there is a set of constants which are naturally involved in the theory of multiple Gamma functions (see [1,15,36] and [57, p. 128]). Some of these constants are introduced here for our latter use. First of all, A denotes the Glaisher–Kinkelin constant defined by (see [9])
"
# 2 n n 1 n2 ffi 1:282427130 . . . : log n þ log A ¼ lim k log k þ þ n!1 2 2 12 4 k¼1 n X
ð1:6Þ
The constants B and C are analogous to the Glaisher–Kinkelin constant A and are defined by (see [26])
" # 3 n X n n2 n n3 7 2 log n þ log B ¼ lim k log k þ þ n!1 3 2 6 9 12 k¼1
ð1:7Þ
and
"
# 4 n X n n3 n2 1 n4 n2 3 ; log n þ log C ¼ lim k log k þ þ n!1 4 2 4 120 16 12 k¼1
ð1:8Þ
respectively. The approximate numerical values of the constant B and C are given by
B ffi 1:03091675
and C ffi 0:97955746 . . . :
The constants A; B and C are also known to be expressible as follows:
log A ¼
1 f0 ð1Þ; 12
log B ¼ f0 ð2Þ;
and
log C ¼
11 f0 ð3Þ; 720
ð1:9Þ
in terms of special values of the derivative of the Riemann zeta function fðsÞ defined by
fðsÞ :¼
8 1 1 X X > > 1 1 1 > s ¼ 12s > n ð2n1Þs < n¼1
RðsÞ > 1 ;
n¼1
1 X > > ð1Þn1 1 > > 1s : 12 ns
RðsÞ > 0; s – 1 :
ð1:10Þ
n¼1
The Riemann Zeta function fðsÞ is a special case of the Hurwitz (or generalized) Zeta function fðs; aÞ defined by
fðs; aÞ :¼
1 X k¼0
1 ðk þ aÞs
RðsÞ > 1; a 2 C n Z0 ;
ð1:11Þ
each of which can be continued meromorphically to the whole complex s-plane except for a simple pole at s ¼ 1 with its residue 1 (see, for details, [57, pp. 88–103]). Remark 1. Adamchik [1] (see also [15] and [57, p. 128]) presented a set of mathematical constants which include the abovedefined A; B and C as special cases. The Polygamma functions wðnÞ ðzÞ ðn 2 NÞ are defined by
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354 nþ1
wðnÞ ðzÞ :¼
d
nþ1
dz
flog CðzÞg ¼
n d n 2 N0 ; z 2 C n Z0 ; n fwðzÞg dz
10345
ð1:12Þ
where N is the set of positive integers and N0 :¼ N [ f0g. In terms of the Hurwitz (or generalized) Zeta function fðs; aÞ, we can write
wðnÞ ðzÞ ¼ ð1Þnþ1 n!
1 X
1
k¼0
ðk þ zÞnþ1
¼ ð1Þnþ1 n! fðn þ 1; zÞ ðn 2 N; z 2 C n Z0 Þ;
ð1:13Þ
which has turned out to be useful in certain applications (see, e.g., [28]). Here, in this paper, we aim at presenting some two-sided inequalities for the multiple Gamma functions Cn for (especially) n ¼ 2; 3; 4; 5. We employ and extend the method initiated by Batir and Cancan [14] and express log Cn ð1 þ xÞ as series involving the Zeta functions (see, e.g., [7,21,24–26,33,53,55–57]). We also derive a more convenient and potentially useful explicit form of the multiple Gamma functions Cn ðn 2 NÞ. The main two-sided inequalities for the multiple Gamma functions Cn ðn ¼ 2; 3; 4; 5Þ (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions Cn when n P q6. 2. Multiple gamma functions There are two known ways to define the n-ple Gamma functions Cn . First of all, Barnes [12] (see also Vardi [59]) defined Cn by using the n-ple Hurwitz Zeta functions (see, e.g., [22], [57, Chapter 2]). Secondly, a recurrence relation of the Weierstrass canonical product forms of the n-ple Gamma functions Cn was given by Vignéras [60] who used the theorem of Dufresnoy and Pisot [34] which provides the existence, uniqueness, and expansion of the series of Weierstrass satisfying a certain functional equation. By making use of the aforementioned Dufresnoy–Pisot theorem and starting with
f1 ðxÞ ¼ c x þ
1 h X x
x i log 1 þ ; n n
n¼1
ð2:1Þ
Vignéras [60] obtained a recurrence relation of Cn ðn 2 NÞ which is stated here as Theorem 1 below. Theorem 1. The n-ple Gamma functions Cn are defined by
Cn ðzÞ ¼ ½Gn ðzÞð1Þ
n1
ðn 2 NÞ;
ð2:2Þ
where
Gn ðz þ 1Þ ¼ exp ½fn ðzÞ
ð2:3Þ
and the functions fn ðzÞ are given by
fn ðzÞ ¼ z An ð1Þ þ
n1 i X pk ðzÞ h ðkÞ fn1 ð0Þ AnðkÞ ð1Þ þ An ðzÞ; k! k¼1
ð2:4Þ
with
An ðzÞ ¼
X m2Nn1 N 0
" n n1 # 1 z 1 z z z n1 n ; þ ð1Þ log 1 þ þ þ ð1Þ n LðmÞ n 1 LðmÞ LðmÞ LðmÞ
ð2:5Þ
where
LðmÞ ¼ m1 þ m2 þ m3 þ þ mn ; if
m ¼ ðm1 ; m2 ; m3 ; ; mn Þ 2 Nn1 N 0 and the polynomials pn ðzÞ given by
8 n n n n < 1 þ 2 þ 3 þ þ ðN 1Þ pn ðzÞ :¼ Bnþ1 ðzÞ Bnþ1 : nþ1
satisfy the following relations:
p0n ðzÞ ¼
B0nþ1 ðzÞ ¼ Bn ðzÞ and pn ð0Þ ¼ 0; nþ1
ðz ¼ N; N 2 N n f1gÞ; ðz 2 CÞ;
ð2:6Þ
Author's personal copy 10346
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
Bn ðzÞ being the Bernoulli polynomial of degree n in z. By analogy with the Bohr–Mollerup theorem (see [8, p. 14]; see also [57, p. 13]), which guarantees the uniqueness of the Gamma function C, one can give, for the double Gamma function and (more generally) for the multiple Gamma functions of order n ðn 2 NÞ, a definition of Artin [8] by means of the following theorem (see Vignéras [60, p. 239]). Theorem 2. For all n 2 N, there exists a unique meromorphic function Gn ðzÞ satisfying each of the following properties: (a) Gn ðz þ 1Þ ¼ Gn1 ðzÞ Gn ðzÞ ðz 2 CÞ; (b) Gn ð1Þ ¼ 1; (c) For x=1; Gn ðxÞ are infinitely differentiable and nþ1
d
nþ1
dx
flog Gn ðxÞg=0;
(d) G0 ðxÞ ¼ x. It is not difficult to verify (see, e.g., [57, pp. 40–41]) that fCn ðzÞg1 is an entire function with zeros at z ¼ k ðk 2 N0 Þ with multiplicities
nþk1
ðn 2 N; k 2 N0 Þ:
n1
ð2:7Þ
In our earlier investigations, we gave explicit forms of the multiple Gamma functions Cn ðn ¼ 3; 4; 5Þ (see, e.g., [21,33]). Now, by observing Eq. (2.7), we can present the following explicit form of the multiple Gamma functions Cn ðn 2 NÞ for a potential and easier future use. Theorem 3. The n-ple Gamma functions Cn in Theorem 1 can be written in a more explicit form as follows:
8 9 nþk2 > " !#> > > 1 > k j n1 k > j¼1 k¼1 > : ;
ð2:8Þ
where Q n ðzÞ is a polynomial in z of degree n given by
"
n1
Q n ðzÞ :¼ ð1Þ
# n1 X pk ðzÞ ðkÞ ðkÞ z An ð1Þ þ fn1 ð0Þ An ð1Þ ; k! k¼1
fn ðzÞ :¼ z An ð1Þ þ
An ðzÞ :¼
1 X k¼1
ð1Þ
n1 i X pk ðzÞ h ðkÞ fn1 ð0Þ AnðkÞ ð1Þ þ An ðzÞ; k! k¼1
n1
nþk2 n1
# " n z X ð1Þj1 zj log 1 þ þ j k j k j¼1
ð2:9Þ
ð2:10Þ
ð2:11Þ
and
pn ðzÞ ¼
nþ1 nþ1 1 X Bnþ1k zk n þ 1 k¼1 k
ðn 2 NÞ:
ð2:12Þ
Remark 2. In order to get explicit forms of the multiple Gamma functions Cn ð1 þ zÞ in Theorem 3, it is indispensable to compute An ð1Þ explicitly. In fact, by using the Taylor–Maclaurin expansion of logð1 þ tÞ in Eq. (2.11) and certain series involving Zeta functions, Choi et al. [21] found that
" j j1 n1 X X X j 0 1 fð‘Þ j‘ zjþ1 f ðk; 1 þ zÞ zjk þ ð1Þjþ1 f0 ðjÞ An ðzÞ ¼ sðn 1; jÞ ð1Þk ð1Þ‘ z þ Hj þ c ðn 1Þ! j¼0 j‘ jþ1 k ‘¼0 k¼0 # nj X fðkÞ kþj ; ð2:13Þ ð1Þk z k þj k¼2 where sðn; kÞ denotes the Stirling numbers of the first kind (see [57, pp. 56–57]) and Hn denotes the harmonic numbers given by
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
Hn :¼
n X 1 k¼1
k
10347
ðn 2 NÞ:
Now, by applying Eq. (2.13) in Theorem 3, we can give explicit forms of the multiple Gamma functions Cn ðn 2 NÞ whose cases ðn ¼ 3; 4; 5Þ are recalled here as the following corollary (see [21]). Corollary 1. Each of the following expressions holds true:
8 9 kþ1 > > > >
1 > k k 2k 2 3k > k¼1 > : ;
ð2:14Þ
where
1 1 c 1 p2 c þ logð2pÞ þ ; c3 ¼ ; 8 4 4 36 6 4 8 9 kþ2 > > > >
1 < = Y kþ2 z z z2 z3 z4 2 3 4 3 C4 ð1 þ zÞ ¼ exp d1 z þ d2 z þ d3 z þ d4 z 1þ 2þ 3 4 ; exp > > k k 2k 3 3k 4k > k¼1 > : ;
c1 ¼
3 1 logð2pÞ log A; 8 4
c2 ¼
ð2:15Þ
where
d1 ¼
7 1 1 log A log B logð2pÞ; 24 2 6
2 c 1 p2 d3 ¼ ; logð2pÞ 9 6 12 54
d2 ¼
1 c 1 1 þ þ logð2pÞ þ log A; 144 6 4 2
11 c p2 fð3Þ þ þ þ ; 144 24 48 12 8 9 kþ3 > > > >
1 < = 2 3 4 5 Y kþ3 z z z z z z 2 3 4 5 4 C5 ð1 þ zÞ ¼ exp e1 z þ e2 z þ e3 z þ e4 z þ e5 z 1þ 2þ 3 4þ 5 ; exp > > k k 2k 3k 4k 5k 4 > k¼1 > : ; d4 ¼
ð2:16Þ where
e1 ¼
409 1 11 1 3 fð3Þ 1 1 log A log C logð2pÞ þ fð4Þ fð5Þ; 1728 8 12 6 16 p2 20 20
e2 ¼
1 c 11 3 fð3Þ þ þ ; logð2pÞ þ log A þ 16 8 48 4 16 p2
e3 ¼
149 11 1 1 1 c logð2pÞ log A fð2Þ; 864 72 8 6 12
e4 ¼
7 1 1 11 1 þ c þ logð2pÞ þ fð2Þ þ fð3Þ; 64 16 48 96 16
e5 ¼
5 1 1 11 1 c fð2Þ fð3Þ fð4Þ: 288 120 20 120 20
3. Use of the method based upon Taylor’s formula By applying Taylor’s formula (see, e.g., [62, p. 209, Theorem 7.44]) to the function hðtÞ given by
hðtÞ ¼ t log t
ðt 2 ½k; k þ xÞÞ
up to its third derivative, Batir and Cancan [14] found that
x x x2 x3 log 1 þ ¼ 2þ k k 2k 3ðk þ lðkÞÞ3 where
ð0 < lðkÞ < xÞ;
lðkÞ can be easily expressed as follows:
ð3:1Þ
Author's personal copy 10348
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
lðkÞ ¼
13 3 x 3 3 log 1 þ þ k: x3 k k x2 2k2 x
Batir and Cancan [14] proved that
ð3:2Þ
lðkÞ is strictly increasing for all k=1 and x > 0 and that
x lð1Þ ¼ lim lðkÞ ¼ : k!1 4 By using these observations, Batir and Cancan [14] derived a two-sided inequality for the double Gamma function G ¼ 1=C2 . More generally, by applying Taylor’s formula to the same function as above to get the remainder on ½k; k þ xÞ of the nth derivative ðn 2 NÞ, we obtain n1 x X ð1Þj1 xj ð1Þn1 xn log 1 þ ;þ ¼ n k k j n k þ ln ðk; xÞ j¼1
ð3:3Þ
k=1; x > 0; n 2 N; 0 < ln ðk; xÞ < x ;
ln ðk; xÞ can be written in the following explicit form:
which shows that
1
ln ðk; xÞ ¼ fg n ðk; xÞgn k ðk=1; x > 0; n 2 NÞ;
ð3:4Þ
where, for convenience,
" # n1 ð1Þn1 n x X ð1Þj1 xj : g n ðk; xÞ :¼ log 1 þ xn k k j j¼1 We now provide some properties of
ð3:5Þ
ln ðk; xÞ as asserted by the following lemmas.
Lemma 1. The following formula holds true:
lim ln ðk; xÞ ¼
k!1
x nþ1
ðx > 0; n 2 NÞ:
ð3:6Þ
Proof. In view of the Taylor–Maclaurin series expansion of logð1 þ tÞ, we find, for any bounded real number x, that
n1 1 x X ð1Þj1 xj X ð1Þj1 xj ð1Þn1 xn ð1Þn xnþ1 1 ðk ! 1Þ; log 1 þ ¼ ¼ þ O nþ2 n þ k k k j j n n þ 1 knþ1 k k j¼1 j¼n so that
ln ðk; xÞ ¼
1n
1n 1 nx 1 1 nx 1 1 þ O k ¼ k 1 þ O k ðk ! 1Þ: n nþ1 nþ2 2 nþ1 k nþ1 k k k k
Using the generalized binomial theorem, we get
ln ðk; xÞ ¼ k
1 X
1n j
j¼0
!
j nx 1 1 þO 2 k; nþ1 k k
where the generalized binomial coefficient
a j
¼
8 0.
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
Lemma 3. The function
10349
ln ðk; xÞ is strictly increasing on k 2 ½1; 1Þ and x > 0 for all n 2 N.
Proof. In view of Eq. (3.5), by considering the sum formula of a finite geometric series, we get
@ n fg ðk; xÞg ¼ n @k n k ðk þ xÞ
ðn 2 NÞ;
which readily yields
@ 1 nþ1 fl ðk; xÞg ¼ n fg n ðk; xÞg n 1: @k n k ðk þ xÞ
ln ðk; xÞ is strictly increasing on k 2 ½1; 1Þ and x > 0 if and only if
It is thus observed that the function
@ k 2 ½1; 1Þ; x > 0 ; fln ðk; xÞg > 0 @k that is, if and only if
( ð1Þ
n1
#)nþ1 n " n1 k x X ð1Þj1 xj 1 n log 1 þ 0 : Here, by setting kx ¼ t in Eq. (3.7), it is seen that the function ln ðk; xÞ is strictly increasing on k 2 ½1; 1Þ and x > 0 if and only if nþ1
Hn ðtÞ :¼ ½hn ðtÞ
t nðnþ1Þ < 0 ðt > 0; n 2 NÞ; nnþ1 ð1 þ tÞn
ð3:8Þ
where, for convenience,
hn ðtÞ :¼ ð1Þ
n1
logð1 þ tÞ
n1 X ð1Þj1 j¼1
j
! t
j
ðt > 0Þ:
We note that
tn1 1þt
0
hn ðtÞ ¼
ðt > 0; n 2 NÞ:
It is observed that, for the inequality in Eq. (3.8), it is equivalent to prove that
Gn ðtÞ :¼ hn ðtÞ
tn n
n ð1 þ tÞnþ1
< 0 ðt > 0; n 2 NÞ:
ð3:9Þ
We now want to prove the following inequality:
G0n ðtÞ ¼
t n1 tn t n1 þ ðt > 0; n 2 NÞ; n n < 0 1þ 1 þ t ðn þ 1Þ ð1 þ tÞ nþ1 ð1 þ tÞnþ1
if and only if
1þ
t
1
n
ðn þ 1Þ ð1 þ tÞnþ1
ð1 þ tÞnþ1 < 0 ðt > 0; n 2 NÞ;
that is, if and only if n
ð1 þ tÞnþ1 þ
t 0; n 2 NÞ;
that is, if and only if n
ð1 þ tÞnþ1 < 1 þ
n t nþ1
ðt > 0; n 2 NÞ;
the last inequality of which holds true by means of the first part of the Bernoulli inequality in Lemma 2. Thus Gn ðtÞ is strictly decreasing on t 2 ð0; 1Þ for all n 2 N, and so we have
Gn ðtÞ < Gn ð0Þ ¼ 0 ðt > 0; n 2 NÞ: This completes the proof of the inequality in Eq. (3.9).
h
Author's personal copy 10350
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
From Lemmas 1 and 3, it is easy to deduce Lemma 4 below. Lemma 4. The following two-sided inequality holds true:
ln ð1; xÞ 5 ln ðk; xÞ
0; n 2 N :
ð3:10Þ
For our convenience and later use, we define the following two functions:
an ðxÞ :¼ 1 þ
x nþ1
and bn ðxÞ :¼ 1 þ ln ð1; xÞ ðx > 0; n 2 NÞ:
ð3:11Þ
We are now ready to present our proposed two-sided inequalities for the multiple Gamma functions Cn ð1 þ xÞ ðn ¼ 2; ; 5; x > 0Þ. Theorem 4. Each of the following two-sided inequalities holds true:
3 3 x x x x ð2pÞ2 exp ½p2 ðxÞ exp qb3 ðxÞ 5 C2 ð1 þ xÞ 5 ð2pÞ2 exp ½p2 ðxÞ exp qa3 ðxÞ ; 3 3
ð3:12Þ
where
x 1 þ ðc þ 1Þx2 ; 2 2 qa3 ðxÞ :¼ f 2; a3 ðxÞ þ ½1 a3 ðxÞ f 3; a3 ðxÞ ; p2 ðxÞ :¼
qb3 ðxÞ :¼ f 2; b3 ðxÞ þ ½1 b3 ðxÞ f 3; b3 ðxÞ ; 4 4 x x exp ½p3 ðxÞ exp q ðxÞ 5 C3 ð1 þ xÞ 5 exp ½p3 ðxÞ exp q ðxÞ ; 8 a4 8 b4
ð3:13Þ
where
p3 ðxÞ :¼ c1 x þ c2 x2 þ c3 x3 ;
qa4 ðxÞ :¼ f 2; a4 ðxÞ þ ½1 2a4 ðxÞ f 3; a4 ðxÞ þ a24 ðxÞ a4 ðxÞ f 4; a4 ðxÞ ;
qb4 ðxÞ :¼ f 2; b4 ðxÞ þ ½1 2b4 ðxÞ f 3; b4 ðxÞ þ b24 ðxÞ b4 ðxÞ f 4; b4 ðxÞ ; x5 x5 exp ½p4 ðxÞ exp qb5 ðxÞ 5 C4 ð1 þ xÞ 5 exp ½p4 ðxÞ exp qa5 ðxÞ ; 30 30
ð3:14Þ
where
p4 ðxÞ :¼ d1 x þ d2 x2 þ d3 x3 þ d4 x4 ;
qa5 ðxÞ :¼ f 2; a5 ðxÞ þ ½6 3a5 ðxÞ f 3; a5 ðxÞ þ 11 12 a5 ðxÞ þ 3 a25 ðxÞ f 4; a5 ðxÞ þ ½1 a5 ðxÞ½2 a5 ðxÞ½3 a5 ðxÞ f 5; a5 ðxÞ ; qb5 ðxÞ :
¼ f 2; b5 ðxÞ þ ½6 3b5 ðxÞ f 3; b5 ðxÞ þ 11 12 b5 ðxÞ þ 3 b25 ðxÞ f 4; b5 ðxÞ þ ½1 b5 ðxÞ½2 b5 ðxÞ½3 b5 ðxÞ f 5; b5 ðxÞ ;
exp ½p5 ðxÞ exp
6 x6 x qa6 ðxÞ 5 C5 ð1 þ xÞ 5 exp ½p5 ðxÞ exp qb6 ðxÞ ; 144 144
where
p5 ðxÞ :¼ e1 x þ e2 x2 þ e3 x3 þ e4 x4 þ e5 x5 ;
ð3:15Þ
Author's personal copy J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
10351
qa6 ðxÞ :
¼ f 2; a6 ðxÞ þ ½10 4a6 ðxÞ f 3; a6 ðxÞ þ 35 30 a6 ðxÞ þ 6 a26 ðxÞ f 4; a6 ðxÞ
þ 50 70 a6 ðxÞ þ 30 a26 ðxÞ 4 a36 ðxÞ f 5; a6 ðxÞ þ ½1 a6 ðxÞ½2 a6 ðxÞ½3 a6 ðxÞ½4 a6 ðxÞ f 5; a6 ðxÞ ;
qb6 ðxÞ :
¼ f 2; b6 ðxÞ þ ½10 4b6 ðxÞ f 3; b6 ðxÞ þ 35 30 b6 ðxÞ þ 6 b26 ðxÞ f 4; b6 ðxÞ
þ 50 70 b6 ðxÞ þ 30 b26 ðxÞ 4 b36 ðxÞ f 5; b6 ðxÞ þ ½1 b6 ðxÞ½2 b6 ðxÞ½3 b6 ðxÞ½4 b6 ðxÞ f 5; b6 ðxÞ :
Proof. Upon taking logarithms on each of the explicit Weierstrass canonical product forms Eqs. (1.3), (2.14), (2.15) and (2.16) of the multiple Gamma functions and applying the result in Lemma 4, if we make a little simplification, we are led to the inequalities asserted by Theorem 4. h Remark 3. In view of the relation Eq. (1.13), the Hurwitz (or generalized) Zeta function fðs; aÞ in Theorem 4 can be replaced by the Polygamma functions wðnÞ ðzÞ as it was done in [14]. 4. Use of series involving the zeta functions Ever since the Goldbach theorem of 1729 (see [53]; see also [57, p. 142]), closed-form evaluations of series involving the Zeta functions have not only attracted many important investigations (see, e.g., [7,21,24–26,33,53,55,57]), but have also found their applications in various ways (see, e.g., [21,25,30]). Our very recent paper [32] deals extensively with the familiar family of the Goldbach–Euler series; in fact, it was motivated essentially by the various developments emerging from the aforementioned Goldbach theorem of 1729. In light of Eq. (1.10), if we take logarithms of both sides in Eqs. (1.3), (2.14), (2.15) and (2.16) and apply the TaylorMaclaurin expansion of logð1 þ tÞ to each of the resulting identities, we find the series representations given by Theorem 5 below. Theorem 5. Each of the following series representations holds true:
log C2 ð1 þ zÞ ¼ c2;1 z þ c2;2 z2 þ
1 X ð1Þkþ1 k¼2
kþ1
log C3 ð1 þ zÞ ¼ c3;1 z þ c3;2 z2 þ c3;3 z3 þ
fðkÞ zkþ1 ;
1 1 1X ð1Þkþ1 1X ð1Þk fðkÞ zkþ1 þ fðkÞ zkþ2 ; 2 k¼2 k þ 1 2 k¼2 k þ 2
log C4 ð1 þ zÞ ¼ c4;1 z þ c4;2 z2 þ c4;3 z3 þ c4;4 z4 þ þ
1 1X ð1Þkþ1 fðnÞzkþ3 ; 6 k¼2 k þ 3
ð4:3Þ 1 1 1X ð1Þkþ1 11 X ð1Þk fðkÞzkþ1 þ fðkÞzkþ2 4 k¼2 k þ 1 24 k¼2 k þ 2
1 1 1X ð1Þkþ1 1 X ð1Þk fðkÞzkþ3 þ fðkÞzkþ4 ; 4 k¼2 k þ 3 24 k¼2 k þ 4
where
c2;1 ¼
1 ½1 logð2pÞ; 2
c3;2 ¼
1 1 c þ logð2pÞ þ ; 8 4 4
c4;1 ¼
7 1 1 logð2pÞ log A log B; 24 6 2
c2;2 ¼
ð4:2Þ
1 1 1X ð1Þkþ1 1X ð1Þk fðkÞ zkþ1 þ fðkÞ zkþ2 3 k¼2 k þ 1 2 k¼2 k þ 2
log C5 ð1 þ zÞ ¼ c5;1 z þ c5;2 z2 þ c5;3 z3 þ c5;4 z4 þ c5;5 z5 þ þ
ð4:1Þ
1 ðc þ 1Þ; 2
c3;1 ¼
3 1 logð2pÞ log A; 8 4
1 c c3;3 ¼ ; 4 6 c4;2 ¼
1 c 1 1 þ þ logð2pÞ þ log A; 144 6 4 2
ð4:4Þ
Author's personal copy 10352
J. Choi, H.M. Srivastava / Applied Mathematics and Computation 219 (2013) 10343–10354
2 1 c c4;3 ¼ logð2pÞ ; 9 12 6
c4;4 ¼
c5;1 ¼ e1 ;
149 11 1 1 c logð2pÞ log A; 864 72 8 6
c5;4 ¼
c5;2 ¼ e2 ;
c5;3 ¼
7 c 1 þ þ logð2pÞ; 64 16 48
11 c þ ; 144 24
5 c þ : 288 120
c5;5 ¼
We next recall here the well-known property of an alternating series of real numbers. Lemma 5. Let fak g1 k¼1 be a decreasing sequence of positive real numbers with ak # 0 as k ! 1. Then the following inequalities hold true: 2n X
ð1Þkþ1 ak
