ON MULTI-ORDER LOGARITHMIC POLYNOMIALS AND THEIR EXPLICIT FORMULAS, RECURRENCE RELATIONS,
AND INEQUALITIES
FENG QI
Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; Department of Mathematics, College of Science, Tianjin
Polytechnic University, Tianjin City, 300387, China
Abstract. In the paper, the author introduces the notions “multi-order log-arithmic numbers” and “multi-order loglog-arithmic polynomials”, establishes an explicit formula, an identity, and two recurrence relations by virtue of the Fa`a di Bruno formula and two identities of the Bell polynomials of the sec-ond kind in terms of the Stirling numbers of the first and secsec-ond kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.
1. Multi-order logarithmic polynomials
Letg(t) =et−1 fort∈
Rand denotexm= (x1, x2, . . . , xm−1, xm) for xk ∈R and 1≤k≤m. Recently, the quantitiesQm,n(xm) were defined by
G(t;xm) = exp(x1g(x2g(· · ·xm−1g(xmg(t))· · ·))) =
∞
X
n=0
Qm,n(xm)
tn
n!
and were called the Bell-Touchard polynomials [22]. Whenm= 1 andx1= 1, the
quantitiesQ1,n(1) =Bnwere called the Bell numbers [1, 5, 8, 17, 18] or exponential numbers [2] and were generalized and applied [1]. When m = 1 andx1 =x is a
variable, the quantities Q1,n(x) =Bn(x) = Tn(x) were called the Bell polynomi-als [20, 21], the Touchard polynomipolynomi-als [19, 22], or exponential polynomipolynomi-als [3, 4, 7] and were applied [9, 10, 11, 12, 19]. In the paper [22], explicit formulas, recurrence relations, determinantal inequalities, product inequalities, logarithmic convexity, logarithmic concavity, and applications of Qm,n(x) were investigated. For more information on this topic, please refer to [22] and closely related references therein.
E-mail address:[email protected], [email protected].
2010 Mathematics Subject Classification. Primary 11B83; Secondary 11A25, 11B73, 11C08, 11C20, 15A15, 26A24, 26A48, 26C05, 26D05, 33B10, 34A05.
Key words and phrases. multi-order logarithmic number; multi-order logarithmic polynomial; explicit formula; identity; recurrence relation; inversion theorem; Bell polynomial of the second kind; Stirling number; determinantal inequality; product inequality; completely monotonic func-tion; logarithmic convexity; Fa`a di Bruno formula.
This paper was typeset usingAMS-LATEX.
1
According to the monograph [6, pp. 140–141], logarithmic polynomials Ln was defined by
ln
∞
X
n=0 gn
tn n!
!
=
∞
X
n=1
Ln
tn n!,
whereg0=G(a) = 1,gn=G(n)(a) forn∈N, andG(x) is an infinitely differentiable function atx=a. In other words, the logarithmic polynomials Ln are expressions for thenth derivative of lnG(x) at the pointx=a. See also [23, Section 5.2].
We now introduce two new notions order logarithmic numbers” and “multi-order logarithmic polynomials”.
Definition 1.1. Forxk ∈Rand 1≤k≤m, denote xm= (x1, x2, . . . , xm−1, xm). Leth(t) = ln(1 +t) fort >−1. DefineH(t;xm) andLm,n(xm) by
H(t;xm) = ln(1 +x1ln(1 +x2ln(1 +· · ·+xm−1ln(1 +xmln(1 +t))· · ·)))
=h(x1h(x2h(· · ·xm−1h(xmh(t))· · ·))) =
∞
X
n=0
Lm,n(xm)
tn
n!.
(1.1)
We call Lm,n(xm) higher order logarithmic polynomials, logarithmic polynomials of orderm,m-variate logarithmic polynomials, multivariate logarithmic polynomi-als, logarithmic polynomials ofmvariablesx1, x2, . . . , xm, multi-order logarithmic polynomials alternatively. When x1 = x2 = · · · = xm−1 = xm = 1, we denote
Lm,n(1, . . . ,1) by Lm,n and call them higher order logarithmic numbers, logarith-mic numbers of orderm, and multi-order logarithmic numbers alternatively.
It is not difficult to see that each of the generating functions G(t;xm) and
H(t;xm) is an inverse function of another one. Since the power series
ln(1 +t) =
∞
X
n=1
(−1)n−1t n
n = ∞
X
n=1
(−1)n−1(n−1)!t n
n!,
we writeL0,0= 0 andL0,n= (−1)n−1(n−1)! forn≥1.
From the limit limt→0H(t;xm) = 0, it follows thatLm,0(xm) = 0 for m≥1. By the softwareMathematica, we can obtain
ln[1 + ln(1 +t)] =t−t2+7t
3
6 −
35t4
24 + 19t5
10 − 917t6
360 +
8791t7
2520 − · · · and
ln[1 +xln(1 +t)] =xt−1
2x(1 +x)t
2+1
6x 2x
2+ 3x+ 2
t3
−1
24x 6x
3+ 12x2+ 11x+ 6
t4+ 1 60x 12x
4+ 30x3+ 35x2+ 25x+ 12
t5
− 1
360x 60x
5+ 180x4+ 255x3+ 225x2+ 137x+ 60
t6+· · ·
which imply that the first few 1-order logarithmic numbersL1,nfor 1≤n≤7 are 1, −2, 7, −35, 228, −1834, 17582
and that the first few 1-order logarithmic polynomialsL1,n(x) for 1≤n≤6 are
x, −x(1 +x), x 2 + 3x+ 2x2
, −x 6 + 11x+ 12x2+ 6x3
,
x 12 + 25x+ 35x2+ 30x3+ 12x4
−x 60 + 137x+ 225x2+ 255x3+ 180x4+ 60x5.
Basing on the above concrete examples, we guess that all multi-order logarith-mic numbers (−1)n−1L
m,n are positive integers and that all multi-order logarith-mic polynomials (−1)n−1L
m,n(xm) are positive integer polynomials of variables
x1, x2, . . . , xmwith degree m×n form, n∈N. This guess will be verified in next section.
2. Recurrence relations and explicit formulas
In this section, we present two recurrence relations, one explicit formula, and one identity for multi-order logarithmic polynomialsLm,n(xm).
Theorem 2.1. Form, n∈N, multi-order logarithmic polynomials Lm,n(xm) can be computed by the forward recurrence relation
Lm,n(xm) = n X
k=0
s(n, k)Lm−1,k(xm−1)xkm, (2.1)
by the backward recurrence relation
Lm−1,n(xm−1)xnm= n X
k=0
S(n, k)Lm,k(xm), (2.2)
by the explicit formula
Lm,n(xm) = "m
Y
q=1
`q−1 X
`q=1
s(`q−1, `q)x `q m−q+1
#
s(`m,1), (2.3)
and by the identity
"m Y
q=1
1
x`q−1 q
`q−1 X
`q=1
S(`q−1, `q) #
Lm,`m(xm) =s(n,1) (2.4)
for `0 = n, where s(n, k) and S(n, k) for n ≥ k ≥ 0 respectively stand for the
Stirling numbers of the first and second kinds which can be generated respectively by
[ln(1 +x)]k
k! =
∞
X
n=k
s(n, k)x n
n!, |x|<1 and
(ex−1)k
k! =
∞
X
n=k
S(n, k)x n
n!.
Proof. In combinatorics, the Bell polynomials of the second kind Bn,k(xn−k+1) are
defined by
Bn,k(xn−k+1) =
X
`1,...,`n∈{0}∪N Pn
i=1i`i=n
Pn i=1`i=k
n! Qn−k+1
i=1 `i! n−k+1
Y
i=1
x i
i! `i
forn≥k≥0, see [6, p. 134, Theorem A], and satisfy two identities Bn,k abx1, ab2x2, . . . , abn−k+1xn−k+1
and
Bn,k(0!,1!,2!, . . . ,(n−k)!) = (−1)n−ks(n, k), (2.6) see [6, p. 135, Theorem B], where aand b are any complex numbers. The Fa`a di Bruno formula for higher order derivatives of composite functions can be described in terms of the Bell polynomials of the second kind Bn,k(xn−k+1) by
dn
dtnf ◦q(x) = n X
k=0
f(k)(q(x))Bn,k q0(x), q00(x), . . . , q(n−k+1)(x), (2.7)
see [6, p. 139, Theorem C].
In [24, p. 171, Theorem 12.1], it is stated that, ifbα andak are a collection of constants independent ofn, then
an= n X
α=0
S(n, α)bα if and only if bn= n X
k=0
s(n, k)ak. (2.8)
By Definition 1.1 and the formulas (2.7), (2.5), and (2.6) in sequence, we have
L1,n(x) = lim t→0
∂nH(t;x)
∂tn = limt→0
∂nln[1 +xln(1 +t)]
∂tn
= lim t→0
n X
k=0
∂kln(1 +v) ∂vk Bn,k
x 1 +t,
−x (1 +t)2, . . . ,
(−1)n−kx(n−k)! (1 +t)n−k+1
= lim t→0
n X
k=1
(−1)k−1(k−1)! (1 +v)k
1 1 +t
n
xk(−1)k+nBn,k(0!,1!, . . . ,(n−k)!)
= n X
k=1
(−1)k−1(k−1)!xks(n, k) = n X
k=1
(−1)k−1s(n, k)(k−1)!xk,
wherev=v(t) =xln(1 +t).
Form≥2, the generating functionH(t;xm) can be rewritten as
H(t;xm) =H(xmh(t);xm−1).
Hence, making use of the formulas (2.7), (2.5), and (2.6) in sequence gives
∂nH(t;x m)
∂tn =
n X
k=0
∂kH(u;x m−1)
∂uk Bn,k xmh
0(t), . . . , x
mh(n−k+1)(t)
= n X
k=0
∂kH(u;x m−1)
∂uk x
k mBn,k
1
1 +t, . . . ,
(−1)n−k(n−k)! (1 +t)n−k+1
= n X
k=0
∂kH(u;x m−1)
∂uk x
k m
1
1 +t
n
(−1)n+kBn,k(0!,1!, . . . ,(n−k)!)
= n X
k=0
∂kH(u;x m−1)
∂uk x
k m
1
1 +t
n
s(n, k),
whereu=u(t) =xmh(t) =xmln(1 +t). Further lettingt→0 and considering the definition in (1.1) yield the recurrence relation (2.1).
Computing the recurrence relation (2.1) results in
Lm,n(xm) = n X
`1=0 s(n, `1)
`1 X
`2=0
s(`1, `2)Lm−2,`2(xm−2)x `2 m−1x
= n X
`1=0 s(n, `1)
`1 X
`2=0
s(`1, `2)· · ·
`m−2 X
`m−1=0
s(`m−2, `m−1)L1,`m−1(x1)x `m−1
2 · · ·x
`2 m−1x
`1 m
= n X
`1=0 s(n, `1)
`1 X
`2=0
s(`1, `2)· · ·
`m−2 X
`m−1=0
s(`m−2, `m−1)
`m−1 X
`m=1
×(−1)`m−1s(`
m−1, `m)(`m−1)!x`1mx
`m−1
2 · · ·x
`2 m−1x`m1
= "m
Y
q=1
`q−1 X
`q=1
s(`q−1, `q)x `q m−q+1
#
(−1)`m−1(`
m−1)!.
The explicit formula (2.3) thus follows.
Applying the above mentioned inversion theorem, [24, Theorem 12.1], to the recurrence relation (2.1) arrives at the backward recurrence relation (2.2) readily.
Applying the above mentioned inversion theorem inductively and recursively to the backward recurrence relation (2.2) produces
n X
k=1
(−1)k−1s(n, k)(k−1)!xk1xn2 =L1,n(x1)xn2 =
n X
k=0
S(n, k)L2,k(x2)
= n X
`1=0
S(n, `1)
1
x`1
3
`1 X
k=0
S(`1, k)L3,k(x3) =· · ·
= n X
`1=0
S(n, `1)
1
x`1
3
`1 X
`2=0
S(`1, `2)· · ·
1
x`m−2 m
`m−2 X
`m−1=0
S(`m−2, `m−1)Lm,`m−1(xm).
Consequently,
(−1)n−1(n−1)!xn1 = n X
k=1
S(n, k) 1 xk
2
k X
`1=0
S(k, `1)
1
x`1
3
`1 X
`2=0
S(`1, `2)· · ·
1
x`m−2 m
`m−2 X
`m−1=0
S(`m−2, `m−1)Lm,`m−1(xm)
which can be rearranged as (2.4). This can also be derived from applying the inversion theorem in (2.8) to the explicit formula (2.3). The proof of Theorem 2.1
is complete.
Remark 2.1. When letting m = 1,2,3, the explicit formula (2.3) and the iden-tity (2.4) reduce to
L1,n(x) = n X
`=1
s(n, `)s(`,1)x`, L2,n(x, y) = n X
`=1
s(n, `)y`
` X
k=1
s(`, k)s(k,1)xk,
L3,n(x, y, z) = n X
i=1
s(n, i)zi
i X
j=1
s(i, j)yj
j X
k=1
s(j, k)s(k,1)xk,
and n X
`=1
S(n, `)L1,`(x) =s(n,1)xn, n X
`=1 S(n, `)
y` ` X
k=1
n X
i=1 S(n, i)
yi i X
j=1 S(i, j)
zj j X
k=1
S(j, k)L3,k(x, y, z) =s(n,1)xn
respectively.
Remark 2.2. When takingx1=x2=· · ·=xm= 1 in Theorem 2.1, we can derive two recurrence relations, one explicit formula, and one identity for multi-order logarithmic numbersLm,n.
3. Inequalities
In this section, we will construct some determinantal and product inequalities for multi-order logarithmic polynomialsLm,n(xm) and derive the logarithmic convex-ity and logarithmic concavconvex-ity for the sequences {Lm,n(x)}n≥1 and Lm,nn!(x) n≥1
respectively.
Theorem 3.1. Let q≥1 be a positive integer, let |eij|q denote a determinant of orderq with elementseij, and letxk>0 for1≤k≤m.
(1) If ai for1≤i≤q are non-negative integers, then
Lm,ai+aj+1(xm)
q ≥0 and
(−1)ai+ajLm,ai+aj+1(xm)
q ≥0. (3.1) (2) Ifa= (a1, a2, . . . , aq)andb= (b1, b2, . . . , bq)are non-increasingq-tuples of
non-negative integers such that Pki=1ai ≥ Pk
i=1bi for1 ≤k ≤q−1 and Pq
i=1ai= Pq
i=1bi,P, then
(−1)P q Y
i=1
Lm,ai+1(xm)≥(−1) P
q Y
i=1
Lm,bi+1(xm). (3.2)
Proof. Recall from [15, Chapter XIII], [25, Chapter 1], and [28, Chapter IV] that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies (−1)kf(k)(x)≥0 on I for all k ≥0. Recall also from [25,
p. 21, Definition 3.1] that a nonnegative function f : (0,∞) → R is a Bernstein function if its first derivative f0 is completely monotonic on (0,∞). Item (iii) in [25, p. 28, Corollary 3.8] reads that the composition of two Bernstein functions is still a Bernstein function. Therefore, since the function h(t) = ln(1 +t) is a Bernstein function, when xk >0 for 1≤k≤m, the generating functionH(t;xm) is a Bernstein function and its first derivativeH(t;xm) is a completely monotonic function on [0,∞). By Definition 1.1, we have
Lm,n(xm) = lim t→0
∂nH(t;xm)
∂tn = limt→0
∂n−1Ht0(t;xm)
∂tn−1 , m, n∈N. (3.3)
In [14] and [15, p. 367], it was proved that if f(t) is completely monotonic on [0,∞), then
f(ai+aj)(t)
q ≥0 and
(−1)ai+ajf(ai+aj)(t)
q ≥0. (3.4)
Applying f(t) to the first derivative Ht0(t;xm) in (3.4), taking the limit t →0+, and considering the equation (3.3) give
lim t→0+
Ht0(t;xm)
(ai+aj) t
q =
and
lim t→0+
(−1)
ai+aj
Ht0(t;xm)
(ai+aj) t
q =
(−1)ai+ajLm,ai+aj+1(xm) q ≥0 The determinantal inequalities in (3.1) follow immediately.
In [15, p. 367, Theorem 2], it was stated that iff(t) is a completely monotonic function on [0,∞), then
q Y
i=1
(−1)aif(ai)(t)≥ q Y
i=1
(−1)bif(bi)(t). (3.5)
Applying f(t) to the derivative Ht0(t;xm) in (3.5), taking the limit t → 0+, and considering the equation (3.3) result in
lim t→0+
q Y
i=1
h
(−1)ai(H0
t(t;xm))
(ai) t
i =
q Y
i=1
(−1)aiL
m,ai+1(xm)
≥ lim t→0+
q Y
i=1
h
(−1)bi(H0
t(t;xm))
(bi) t
i =
q Y
i=1
(−1)biL
m,bi+1(xm)
.
The product inequality (3.2) thus follows. Theorem 3.1 is proved.
Corollary 3.1. Let xk >0for1≤k≤m. If`≥0 andq≥k≥0, then (−1)(k+`)q[Lm,q+`+1(xm)]k[Lm,`+1(xm)]q−k ≥(−1)(k+`)q[Lm,k+`+1(xm)]q. Proof. This follows from taking
a= ( k
z }| {
q+`, . . . , q+`,
q−k z }| {
`, . . . , `) and b= (k+`, k+`, . . . , k+`)
in the inequality (3.2). The proof of Corollary 3.1 is complete.
Theorem 3.2. Let xk > 0 for 1 ≤ k ≤ m. Then {(−1)n−1Lm,n(xm)}n≥1 is a
logarithmically convex sequence.
Proof. In [15, p. 369] and [16, p. 429, Remark], it was obtained that if f(t) is a completely monotonic function such thatf(k)(t)6= 0 fork≥0, then the sequence
ln
(−1)k−1f(k−1)(t)
, k≥1 (3.6)
is convex. Applying this result toH0(t;x
m) figures out that the sequence
ln
(−1)k−1(H0(t;xm))
(k−1)
t
→ln
(−1)k−1Lm,k(xm), t→0+ fork ≥1 is convex. Equivalently speaking, the sequence {(−1)n−1L
m,n(xm)}n≥1
is logarithmically convex.
Letting`≥1,n= 2,a1=`+2,a2=`, andb1=b2=`+1 in the inequality (3.2)
alternatively leads to
Lm,`+1(xm)Lm,`+3(xm)≥L2m,`+2(xm) which implies that the positive sequence{(−1)n−1L
m,n(xm)}n≥2is logarithmically
Theorem 3.3. Let xk>0 for1≤k≤m. For q≥0and n∈N, we have "
(−1)(n+1)(q+1) n Y
`=0
Lm,q+2`+2(xm)
#1/(n+1)
≥
"
(−1)nq n−1
Y
`=0
Lm,q+2`+3(xm) #1/n
.
(3.7)
Proof. Iff(t) is a completely monotonic function on (0,∞), then, by the convexity of the sequence (3.6) and Nanson’s inequality listed in [13, p. 205, 3.2.27],
" n Y
`=0
(−1)q+2`+1f(q+2`+1)(t)
#1/(n+1)
≥
" n Y
`=1
(−1)q+2`f(q+2`)(t) #1/n
forq≥0. Replacingf(t) byH0(t;xm) in the above inequality results in " n
Y
`=0
(−1)q+1(H0(t;xm))
(q+2`+1)
t
#1/(n+1)
≥
" n Y
`=1
(−1)q(H0(t;xm))
(q+2`)
t
#1/n
for q ≥ 0. Letting t → 0+ in the above inequality leads to (3.7). The proof of
Theorem 3.3 is complete.
Theorem 3.4. Let xk > 0 for 1 ≤ k ≤ m. If ` ≥ 0, n ≥ k ≥ q, 2k ≥ n, and 2q≥n, then
(−1)nLm,k+`+1(xm)Lm,n−k+`+1(xm)≥(−1)nLm,q+`+1(xm)Lm,n−q+`+1(xm) (3.8) and
(−1)nkLm,k+1(xm) n
≤(−1)nkLm,n+1(xm) k
[Lm,1(xm)]n−k. (3.9)
Proof. In [26, p. 397, Theorem D], it was recovered that, if f(t) is a completely monotonic function on (0,∞) and ifn≥k≥q,k≥n−k, andq≥n−q, then
(−1)nf(k)(t)f(n−k)(t)≥(−1)nf(q)(t)f(n−q)(t). Replacingf(t) by the function (−1)`[H0(t;xm)]
(`)
t in the above inequality leads to
(−1)n[H0(t;xm)]
(k+`)
t [H
0(t;x
m)]
(n−k+`)
t
≥(−1)n[H0(t;xm)]
(q+`)
t [H0(t;xm)]
(n−q+`)
t .
Further takingt→0+ finds the inequality (3.8).
Whenf(t) is a completely monotonic function, the inequality
(−1)nkf(k)(t)n
≤(−1)nkf(n)(t)k
fn−k(t)
is valid for 0< k < n. See [15, p. 368]. SubstitutingH0(t;xm) forf(t) reveals
(−1)nk(H0(t;xm))(k) n
≤(−1)nk(H0(t;xm))(n) k
[H0(t;xm)]n−k.
Lettingt→0+ leads to (3.9). The proof of Theorem 3.4 is complete.
Theorem 3.5. Let xk>0 for1≤k≤m. For `≥0 andq, n∈N, let
Gm,`,q,n =Lm,`+2q+n+1(xm)[Lm,`+1(xm)]2
−Lm,`+q+n+1(xm)Lm,`+q+1(xm)Lm,`+1(xm)
Hm,`,q,n =Lm,`+2q+n+1(xm)[Lm,`+1(xm)]2
−2Lm,`+q+n+1(xm)Lm,`+q+1(xm)Lm,`+1(xm) +Lm,`+n+1(xm)[Lm,`+q+1(xm)]2,
Im,`,q,n =Lm,`+2q+n+1(xm)[Lm,`+1(xm)]2
−2Lm,`+n+1(xm)Lm,`+2q+1(xm)Lm,`+1(xm) +Lm,`+n+1(xm)[Lm,`+q+1(xm)]2.
Then
Gm,`,q,n ≥0, Hm,`,q,n≥0,
Hm,`,q,nQGm,`,q,n when q≶n,
Im,`,q,n≥ Gm,`,q,n≥0 whenn≥q.
(3.10)
Proof. In [27, Theorem 1 and Remark 2], it was obtained that, iff(t) is completely monotonic on (0,∞) and
Gq,n(t) = (−1)nf(n+2q)(t)f2(t)−f(n+q)(t)f(q)(t)f(t)
−f(n)(t)f(2q)(t)f(t) +f(n)(t)
f(q)(t)2
,
Hq,n(t) = (−1)nf(n+2q)(t)f2(t)−2f(n+q)(t)f(q)(t)f(t) +f(n)(t)f(q)(t)
2 ,
Iq,n(t) = (−1)nf(n+2q)(t)f2(t)−2f(n)(t)f(2q)(t)f(t) +f(n)(t)f(q)(t)
2
forn, q∈N, then
Gq,n(t)≥0, Hq,n(t)≥0,
Hq,n(t)QGq,n(t) whenq≶n,
Iq,n(t)≥Gq,n(t)≥0 whenn≥q.
(3.11)
Replacingf(t) by (−1)`[H0(t;x
m)]
(`)
t inGq,n(t),Hq,n(t), andIq,n(t) and simplifying produce
Gq,n(t) = (−1)`+n
[H0(t;xm)]
(`+2q+n)
t
[H0(t;xm)]
(`)
t 2
−[H0(t;xm)]
(`+q+n)
t [H0(t;xm)]
(`+q)
t [H0(t;xm)]
(`)
t
−[H0(t;xm)]
(`+n)
t [H
0(t;x
m)]
(`+2q)
t [H
0(t;x
m)]
(`)
t
+ [H0(t;xm)]
(`+n)
t
[H0(t;xm)]
(`+q)
t 2
,
Hq,n(t) = (−1)`+n
[H0(t;xm)]
(`+2q+n)
t
[H0(t;xm)]
(`)
t 2
−2[H0(t;xm)]
(`+q+n)
t [H0(t;xm)]
(`+q)
t [H0(t;xm)]
(`)
t
+ [H0(t;xm)]
(`+n)
t
[H0(t;xm)]
(`+q)
t 2
,
Iq,n(t) = (−1)`+n[H0(t;xm)]
(`+2q+n)
t
[H0(t;xm)]
(`)
t 2
−2[H0(t;xm)]
(`+n)
t [H
0(t;x
m)]
(`+2q)
t [H
0(t;x
m)]
(`)
t
+ [H0(t;xm)]
(`+n)
t
[H0(t;xm)]
(`+q)
t 2
.
Further takingt→0+ reveals
lim
Substituting these quantities into (3.11) and simplifying bring about inequalities
in (3.10). The proof of Theorem 3.5 is complete.
Remark 3.1. When takingx1 =· · · = xm = 1, all results in this section become conclusions for multi-order exponential numbersLm,n.
References
[1] N. Asai, I. Kubo, and H.-H. Kuo, Bell numbers, log-concavity, and log-convexity, Acta Appl. Math. 63 (2000), no. 1-3, 79–87; Available online at https://doi.org/10.1023/A: 1010738827855.
[2] E. T. Bell,Exponential numbers, Amer. Math. Monthly41(1934), no. 7, 411–419; Available online athttp://dx.doi.org/10.2307/2300300.
[3] E. T. Bell,Exponential polynomials, Ann. of Math. (2)35(1934), no. 2, 258–277; Available online athttp://dx.doi.org/10.2307/1968431.
[4] K. N. Boyadzhiev,Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, Abstr. Appl. Anal. 2009, Art. ID 168672, 18 pp; Available online athttp://dx. doi.org/10.1155/2009/168672.
[5] E. R. Canfield,Engel’s inequality for Bell numbers, J. Combin. Theory Ser. A72(1995), no. 1, 184–187; Available online athttps://doi.org/10.1016/0097-3165(95)90033-0. [6] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised
and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974.
[7] A. Dil and V. Kurt, Investigating geometric and exponential polynomials with Euler-Seidel matrices. J. Integer Seq.14(2011), no. 4, Article 11.4.6, 12 pp.
[8] B.-N. Guo and F. Qi,An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, Glob. J. Math. Anal.2(2014), no. 4, 243–248; Available online at
https://doi.org/10.14419/gjma.v2i4.3310.
[9] W.-X. Ma,Bilinear equations, Bell polynomials and linear superposition principle, J. Phys. Conf. Ser.411(2013), no. 1, Aricle ID 012021, 11 pages; Available online athttps://doi. org/10.1088/1742-6596/411/1/012021.
[10] W.-X. Ma, Bilinear equations and resonant solutions characterized by Bell polynomials, Rep. Math. Phys.72(2013), no. 1, 41–56; Available online at https://doi.org/10.1016/ S0034-4877(14)60003-3.
[11] W.-X. Ma, Trilinear equations, Bell polynomials, and resonant solutions, Front. Math. China 8 (2013), no. 5, 1139–1156; Available online at https://doi.org/10.1007/ s11464-013-0319-5.
[12] F. Mirzaee, Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polynomials, Comput. Methods Differ. Equ.5(2017), no. 2, 88–102.
[13] D. S. Mitrinovi´c,Analytic Inequalities, Springer-Verlag, 1970.
[14] D. S. Mitrinovi´c and J. E. Peˇcari´c,On two-place completely monotonic functions, Anzeiger ¨
Oster. Akad. Wiss. Math.-Natturwiss. Kl.126(1989), 85–88.
[15] D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Classical and New Inequalities in Anal-ysis, Kluwer Academic Publishers, 1993; Available online at https://doi.org/10.1007/ 978-94-017-1043-5.
[16] J. E. Peˇcari´c,Remarks on some inequalities of A. M. Fink, J. Math. Anal. Appl.104(1984), no. 2, 428–431; Available online athttps://doi.org/10.1016/0022-247X(84)90006-4. [17] F. Qi,An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers,
Mediterr. J. Math.13(2016), no. 5, 2795–2800; Available online athttps://doi.org/10. 1007/s00009-015-0655-7.
[18] F. Qi,Some inequalities for the Bell numbers, Proc. Indian Acad. Sci. Math. Sci.127(2017), no. 4, 551–564; Available online athttps://doi.org/10.1007/s12044-017-0355-2. [19] F. Qi,Some properties of the Touchard polynomials, ResearchGate Working Paper (2017),
available online athttps://doi.org/10.13140/RG.2.2.30022.16967.
[21] F. Qi, D.-W. Niu, and B.-N. Guo,Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, Preprints2017, 2017080044, 10 pages; Available online athttps://doi.org/10.20944/preprints201708.0044.v1.
[22] F. Qi, D.-W. Niu, D. Lim, and B.-N. Guo,A unified generalization of the Bell numbers and the Touchard polynomials and its properties, ResearchGate Working Paper (2017), available online athttps://doi.org/10.13140/RG.2.2.36733.05603.
[23] F. Qi, X.-T. Shi, and F.-F. Liu, Expansions of the exponential and the logarithm of power series and applications, Arab. J. Math. (Springer)6(2017), no. 2, 95–108; Available online athttps://doi.org/10.1007/s40065-017-0166-4.
[24] J. Quaintance and H. W. Gould, Combinatorial Identities for Stirling Numbers. The un-published notes of H. W. Gould. With a foreword by George E. Andrews. World Scientific Publishing Co. Pte. Ltd., Singapore, 2016.
[25] R. L. Schilling, R. Song, and Z. Vondraˇcek,Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics37, Walter de Gruyter, Berlin, Germany, 2012; Available online athttps://doi.org/10.1515/9783110269338.
[26] H. van Haeringen,Completely monotonic and related functions, J. Math. Anal. Appl.204
(1996), no. 2, 389–408; Available online athttps://doi.org/10.1006/jmaa.1996.0443. [27] H. van Haeringen,Inequalities for real powers of completely monotonic functions, J. Math.
Anal. Appl.210(1997), no. 1, 102–113; Available online athttps://doi.org/10.1006/jmaa. 1997.5376.
[28] D. V. Widder,The Laplace Transform, Princeton University Press, Princeton, 1946.
