Journal of Inequalities in Pure and
Applied Mathematics
APPLICATIONS OF THE EXTENDED HERMITE-HADAMARD
INEQUALITY
volume 7, issue 1, article 24,
2006.
Z. RETKES
University of Szeged
Bolyai Institute
Aradi vértanúk tere 1
Szeged, H-6720 Hungary.
Received 28 September, 2005;
accepted 13 November, 2005.
Communicated by: P.S. Bullen
EMail: retkes@math.u-szeged.hu
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2000
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Abstract
In this paper we give some combinatorial applications according to a new extension of the classical Hermite-Hadamard inequality proved in [1].
2000 Mathematics Subject Classification: 26D10, 49J40, 49K40.
Key words: Dynamical Systems, Monotone Trajectories, Generalized Jacobian, Variational Inequalities.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Asymptotical Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Formulae for Power Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References
Monotone Trajectories of
Dynamical Systems and
Clarke’s Generalized Jacobian
Giovanni P. Crespi and
mrocca@eco.uninsubria.it
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1.
Introduction
In the paper [1], we have proved the following generalization of the classical Hermite-Hadamard inequality for convex functions, extended it to n nodes:
Suppose that −∞ ≤ a < b ≤ ∞, f : (a, b) → R is a strict convex function,
xi ∈ (a, b), i = 1, . . . , n such that xi 6= xj if 1 ≤ i < j ≤ n. Then the following
inequality holds:
n
X
i=1
n
F (n−1) (xi )
1 X
<
f (xi ).
Πi (x1 , . . . , xn )
n! i=1
In the concave case < sign is changed to >. Here Πi (x1 , . . . , xn ) :=
Qn
k=1
k6=i
Monotone Trajectories of
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(xi −
xk ) and F (0) (s), F (1) (s), . . ., F (n−1) (s), . . . is a sequence of functions defined
d
recursively by F (0) (s) = f (s) and ds
F (n) (s) = F (n−1) (s), n = 1, 2, . . .. These
sequences of functions of f are known as the iterated integrals of f . As an
application of this main result we have proved the following identities:
(1.1)
n
X
k=1
(1.2)
xnk
Πk (x1 , . . . , xn )
n
X
k=1
=
n
X
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xk ,
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xn−1
k
= 1,
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(1.3)
n
n
n
X
Y
X
1
1
n−1
= (−1)
xk
2
xk
x Πk (x1 , . . . , xn )
k=1
k=1
k=1 k
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and
(1.4)
n
n
Y
X
1
1
n−1
= (−1)
xk
xk Πk (x1 , . . . , xn )
k=1
k=1
(xk 6= 0).
In this paper we apply the formulae above to obtain closed combinatorial formulae and to investigate the asymptotic behaviours of these sums. For the
sake of the convenience of the reader we collect the transformational regulations for the quantity Πk (x1 , . . . , xn ) which plays a significant role in all of the
formulae. Note that all of them are simple consequences of the definition of
Πk (x1 , . . . , xn ).
Monotone Trajectories of
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Giovanni P. Crespi and
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a) If α ∈ R, then
Πk (α + x1 , . . . , α + xn ) = Πk (x1 , . . . , xn ),
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that is, Πk is shift invariant.
b)
Πk (αx1 , . . . , αxn ) = αn−1 Πk (x1 , . . . , xn ).
c)
Πk (α − x1 , . . . , α − xn ) = (−1)n−1 Πk (x1 , . . . , xn ).
d) If xk 6= 0, k = 1, . . . , n, then
1
1
Πk (x1 , . . . , xn )
Πk
,...,
= (−1)n−1 n−2 n
.
x1
xn
xk Πj=1 xj
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2.
Asymptotical Formulae
a) Let 1 < x1 < · · · < xn be variables and s1 = x1 , s2 = x1 + x2 , . . .,
sn = x1 + · · · + xn . Then si 6= sj if 1 ≤ i < j ≤ n, hence formula (1.1) can be
applied to s1 , . . . , sn , consequently we have
n
n
X
X
snj
(2.1)
sj =
.
Πj (s1 , . . . , sn )
j=1
j=1
For the left-hand side of (2.1)
n
n
X
X
sj = x1 + (x1 + x2 ) + · · · + (x1 + · · · + xn ) =
j · xn−j+1 .
j=1
j=1
The definition of Πj implies
Πj (s1 , . . . , sn )
= (sj − s1 ) · · · (sj − sj−1 )(sj − sj+1 ) · · · (sj − sn )
= (x2 + · · · + xj ) · · · (xj−1 + xj ) · xj (−1)n−j xj+1 (xj+1 + xj+2 )
· · · (xj+1 + · · · + xn ).
Using the above expressions we have the following identities for all 1 < x1 <
· · · < xn :
(2.2)
n
X
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j · xn−j+1
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j=1
=
Monotone Trajectories of
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n
X
j=1
n−j
n
(−1) (x1 + · · · + xj )
.
(x2 + · · · + xj ) · · · (xj−1 + xj )xj xj+1 · · · (xj+1 + · · · + xn )
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Now, if xk → 1, k = 1, . . . , n then (2.2) gives
X
n
n
X
n+1
(−1)n−j j n
=
j=
.
(j − 1)!(n − j)!
2
j=1
j=1
Multiplying both sides by (n − 1)! leads us to
n
X
n−j n
(−1)
j
j=1
n−1
(n + 1)!
=
.
j−1
2
Monotone Trajectories of
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b) Apply now formula (1.3) to xk = 1 + nk , k = 1, . . . , n. Hence
n
X
k=1
1
1+
k
n
= (−1)n−1
n Y
1+
k=1
= (−1)n−1
= (−1)n−1
1
nn
1
nn
n
Y
k=1
n
Y
k=1
n
Y
k
n
X
n
(n + k)
k=1
n
X
k=1
(n + k)
(1 +
2
(n +
k)2 Π
n
X
k=1
n
X
1
+ n1 , . . . , 1 + nn )
k 2
) Πk (1
n
n
1 2
n
k( n , n , . . . , n )
2
(n +
k)2
n
Π (1, . . . , n)
nn−1 k
1
n
= (−1)
(n + k)
2
(n + k) (k − 1)!(−1)n−k (n − k)!
k=1
k=1
X
n
2n
(−1)k−1 n − 1
.
=n·
n k=1 (n + k)2 k − 1
n−1
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From this equality chain we conclude that
(2.3)
n
n
1X 1
2n X (−1)k−1 n − 1
=
n k=1 (n + k)2 k − 1
n k=1 1 +
Since
n
1X 1
lim
n→∞ n
1+
k=1
Z
k
n
=
0
1
k
n
=
n
X
k=1
1
.
n+k
1
du = log 2,
1+u
Monotone Trajectories of
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Clarke’s Generalized Jacobian
hence the left-hand side of (2.3) is convergent and
lim
n→∞
n
2n X (−1)k−1 n − 1
= log 2.
n k=1 (n + k)2 k − 1
Giovanni P. Crespi and
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n
Pursuant to the Stirling formula 2n
∼ √4πn (n → ∞), consequently we
n
have the following asymptotic expression for the sum on the left:
n
X
k=1
√
√
(−1)k−1 n − 1
n
∼ π log 2 n
2
(n + k) k − 1
4
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(n → ∞).
c) This example follows along the lines above but choosing xk = 1 +
(k = 1, . . . , n). Without the detailed computation we have
Qn
n
n
n
2
2 X
1X
1
(−1)k−1 k 2 k
k=1 (n + k )
.
(2.4)
= 2n
·
2 + k 2 )2 n+k
n k=1 1 + ( nk )2
(n!)2
(n
n
k=1
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k 2
n
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From this equality the limit of the right-hand side exists and
Qn
n
n
n
2
2 X
k−1 2
(n
+
k
)
(−1)
k
1X
1
k
k=1
· n+k = lim
lim 2n ·
2
2
2
2
n→∞
n→∞ n
(n!)
(n + k )
1 + ( nk )2
n
k=1
k=1
Z 1
1
=
du
2
0 1+u
π
= arctan 1 = .
4
Combining these observations we have
n
n
X
(−1)k−1 k 2 k
π
(n!)2
∼
·
(n2 + k 2 )2 n+k
8n Πnk=1 (n2 + k 2 )
n
k=1
(n → ∞).
(n2 + k 2 ) = n2n
k=1
n
Y
k=1
"
2 #
k
1+
.
n
On the other hand
n
Y
1
log
n
k=1
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Moreover (n!)2 ∼ 2πn( ne )2n and
n
Y
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"
"
2 #
2 #
n
k
k
1X
1+
=
log 1 +
n
n k=1
n
Z 1
→
log(1 + x2 )dx (n → ∞),
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hence
n
Y
k=1
"
2 #
R1
k
2
1+
∼ en 0 log(1+x )dx .
n
Using integration by parts to evaluate the integral in the exponent gives
Z 1
π
log(1 + x2 )dx = log 2 + − 2 = α > 0.
2
0
Putting together these results gives the following asymptotical expression:
n
n
X
(−1)k−1 k 2 k
π 2 −n(log 2+ π )
2
∼
e
(n → ∞).
2 + k 2 )2 n+k
(n
4
n
k=1
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3.
Formulae for Power Sums
Let u1 , . . . , un ∈ R such that ui 6= 0, |ui | < 1, ui 6= uj if 1 ≤ i < j ≤ n and
xi = 1 − ui (i = 1, . . . , n). Then applying (1.3) gives
n
X
i=1
n
n
Y
X
1
1
n−1
= (−1)
(1 − uk )
2
1 − ui
(1 − ui ) Πi (1 − u1 , . . . , 1 − un )
i=1
k=1
=
n
Y
(1 − uk )
i=1
k=1
Under the above conditions
(3.1)
n
X
i=1
n
X
(1 − ui
1
1−ui
=
n
∞
)2 Π
P∞
`=0
1
.
i (u1 , . . . , un )
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u`i , hence
∞
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∞
n
XX
XX
X
1
=
u`i =
u`i =
P (`),
1 − ui
i=1 `=0
`=0 i=1
`=0
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where we use the denotation P (`) := u`1 + · · · + u`n for (` = 0, 1, 2, . . .), and
thus we have the following identities for u1 , . . . , un :
(3.2)
∞
X
n
Y
P (`) =
`=0
(1 − uk )
n
X
i=1
k=1
(1 − ui
)2 Π
1
.
i (u1 , . . . , un )
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Apply now (3.2) to uk =
∞
X
`=0
P (`) =
k
n+1
∞ X
n X
`=0 k=1
(k = 1, . . . , n). For the left-hand side:
k
n+1
`
=
∞
X
`=0
1
(n + 1)`
n
X
k=1
k` =
∞
X
`=0
P̃ (`)
,
(n + 1)`
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where P̃ (`) = 1` + 2` + · · · + n` (` = 0, 1, 2, . . .). On the other hand
n Y
k
n!
1−
and
=
n+1
(n + 1)n
k=1
1
n
(−1)n−i (i − 1)!(n − i)!
Πi
,...,
=
.
n+1
n+1
(n + 1)n−1
Putting these expressions into the right-hand side of (3.2) we obtain
n
∞
X
X
(−1)j−1 n − 1
P̃ (`)
= n(n + 1)
.
(3.3)
(n + 1)`
j2
j−1
j=1
`=0
A simple rearrangemant of (3.3) and using
∞
X
`=0
n−1
n
j j−1
=
n
j
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and since it is known that
j=1
j
j
Z
=
0
1
n
X1
1 − (1 − s)n
ds =
,
s
k
k=1
consequently we have the following identity for all fixed n:
∞
X
`=0
n
X1
P̃ (`)
=
.
(n + 1)`+1
k
k=1
Giovanni P. Crespi and
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gives
n
X
P̃ (`)
(−1)j−1 n
,
=
(n + 1)`+1
j
j
j=1
n
X
(−1)j−1 n
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Let us start now with (3.1) substituting uk = x1k , xk > 1 (k = 1, . . . , n) and
xi 6= xj if 1 ≤ i < j ≤ n. Using the same technique applied above and the
transformational rule d) we get the following identity for x1 , . . . , xn :
(3.4)
n
∞ X
n
n
X
X
Y
1
xni
n−1
(x
−
1)
=
(−1)
.
k
(xi − 1)2 Πi (x1 , . . . , xn )
x`
i=1
`=0 k=1 k
k=1
Putting xk = k + 1 (k = 1, . . . , n) in (3.4) and simplifying the right-hand side
we have the following equality:
X
n
∞ X
1
(−1)i−1
1
n n
+ ··· +
=
(i + 1)
.
(3.5)
`
`
2
(n
+
1)
i
i
i=1
`=0
1
[n+1]
Observe that 21` + · · · + (n+1)
(`) − 1, where ζ [n+1] (`) denotes the
` = ζ
P
1
(n + 1)th partial sum of the series ζ(`) = ∞
k=1 k` , if ` ≥ 2. Moreover
Pn 1 let Hn
denote the nth partial sum of the harmonic series, that is Hn = k=1 k . Using
this quantity and separating the summands corresponding to ` = 0 and ` = 1
gives
∞
n
X
X
(−1)i−1
[n+1]
n n
(3.6)
n + Hn+1 − 1 +
[ζ
(`) − 1] =
(i + 1)
.
i
i
i=1
`=2
Rearranging (3.6) and taking the limit as n → ∞ yields that:
" n
#
∞
X
X (−1)i−1
n
(i + 1)n
− n − Hn+1 .
(3.7)
[ζ(`) − 1] = 1 + lim
n→∞
i
i
1=1
`=2
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This relation prompted us to investigate the quantities arising in both sides
of the equality. On one hand the sum of the series is equal to 1 as an easy
computation shows below:
∞
X
∞ X
∞
∞
∞
X
X
1
1 X 1
[ζ(`) − 1] =
=
`
k
k 2 j=0 k j
`=2
`=2 k=2
k=2
∞
X
1 1
=
k2 1 −
k=2
=
∞
X
k=2
Hence
"
lim
n→∞
n
X
(−1)i−1
i=1
i
1
k
1
= 1.
k(k − 1)
#
n
− n − Hn+1 = 0.
(i + 1)n
i
In fact we prove more in the following lemma.
Lemma 3.1. For all n ≥ 1 the identity below holds
n
X
(−1)i−1
i=1
i
n
(i + 1)
= n + Hn .
i
n
Proof. In the paper [1] we proved that if x1 , . . . , xn ∈ R such that xi 6= xj if
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1 ≤ i < j ≤ n then
 Pn

k=1 xk

n
j

X
xi
1
=
Πi (xi , . . . , xn ) 

i=1

0
Applying this result to xi =
1
i
if
j=n
if
j =n−1
if
0 ≤ j ≤ n − 2.
(i = 1, . . . , n) gives that

Hn


n

i−1
X
n
(−1)
1
in−j
=

i
i

i=1

0
if
j=n
if
j =n−1
if
0 ≤ j ≤ n − 2.
Moreover
n
X
i=1
Monotone Trajectories of
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Giovanni P. Crespi and
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" n
# X
n
i−1 X
(−1)i−1
n
(−1)
n
n
(i + 1)n
=
in−k
i
i
i
k
i
i=1
k=0
"
#
n
n
X
X
(−1)i−1 n−k n
n
=
i
i
i
k
k=0 i=1
n
n
=
+ Hn
= n + Hn
n−1
n
as we stated.
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References
[1] Z. RETKES, An extension of the Hermite-Hadamard inequality (submitted)
[2] S.S. DRAGOMIR AND C.E.M. PEARCE, Selected Topics on HermiteHadamard Inequalities, RGMIA Monographs, Victoria University, 2000.
[ONLINE:
http://rgmia.vu.edu.au/monographs/index.
html]
[3] W. RUDIN, Real and Complex Analysis, McGraw-Hill Book Co., 1966.
[4] G.H. HARDY, J.E. LITTLEWOOD
bridge University Press, 1934.
AND
G. PÓLYA, Inequalities, Cam-
Monotone Trajectories of
Dynamical Systems and
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