Journal of Inequalities in Pure and
Applied Mathematics
ON SOME POLYNOMIAL–LIKE INEQUALITIES OF BRENNER
AND ALZER
volume 6, issue 1, article 24,
2005.
C.E.M. PEARCE AND J. PEČARIĆ
School of Applied Mathematics
The University of Adelaide
Adelaide SA 5005
Australia
EMail: cpearce@maths.adelaide.edu.au
URL: http://www.maths.adelaide.edu.au/applied/staff/cpearce.html
Faculty of Textile Technology
University of Zagreb
Pierottijeva 6, 10000 Zagreb
Croatia
EMail: pecaric@mahazu.hazu.hr
URL: http://mahazu.hazu.hr/DepMPCS/indexJP.html
Received 30 September, 2003;
accepted 07 November, 2003.
Communicated by: T.M. Mills
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
135-03
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Abstract
Refinements and extensions are presented for some inequalities of Brenner
and Alzer for certain polynomial–like functions.
2000 Mathematics Subject Classification: Primary 26D15.
Key words: Polynomial inequalities, Switching inequalities, Jensen’s inequality
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Concavity of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
7
C.E.M. Pearce and J. Pečarić
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1.
Introduction
Brenner [2] has given some interesting inequalities for certain polynomial–like
functions. In particular he derived the following.
P
Theorem A. Suppose m > 1, 0 < p1 , . . . , pk < 1 and Pk = ki=1 pi ≤ 1. Then
(1.1)
k
X
m
m
(1 − pm
i ) > k − 1 + (1 − Pk ) .
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
i=1
Alzer [1] considered the sum
Ak (x, s) =
k X
s
i=0
i
C.E.M. Pearce and J. Pečarić
i
s−i
x (1 − x)
(0 ≤ x ≤ 1)
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and proved the following companion inequality to (1.1).
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Theorem B. Let p, q, m and n be positive real numbers and k a nonnegative
integer. If p + q ≤ 1 and m, n > k + 1, then
(1.2)
Ak (pm , n) + Ak (q n , m) > 1 + Ak ((p + q)min(m,n) , max(m, n)).
In the special case k = 0 this provides
(1.3) (1 − pm )n + (1 − q n )m > 1 + (1 − (p + q)min(m,n) )max(m,n)
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for p, q > 0.
In Section 2 we use (1.3) to derive an improvement of Theorem A and a
corresponding version of Theorem B. In Section 3 we give a related Jensen
inequality and concavity result.
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2.
Basic Results
Theorem 2.1. Under the conditions of Theorem A we have
k
X
(2.1)
m
m m
(1 − pm
i ) > k − 1 + (1 − Pk ) .
i=1
Proof. We proceed by mathematical induction, (1.3) with n = m providing a
basis
(2.2) (1−pm )m +(1−q m )m > 1+(1−(p+q)m )m
for p, q > 0 and p + q ≤ 1
for k = 2. For the inductive step, suppose that (2.1) holds for some k ≥ 2, so
that
k+1
X
m
(1 − pm
i ) =
i=1
k
X
C.E.M. Pearce and J. Pečarić
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m
m
m
(1 − pm
i ) + (1 − pk+1 )
Contents
i=1
m
> k − 1 + (1 − Pkm )m + (1 − pm
k+1 ) .
Applying (2.2) yields
k+1
X
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
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m
m
m
(1 − pm
i ) > k − 1 + 1 + (1 − (Pk + pk+1 ) )
i=1
m
= k + 1 − Pk+1
m
.
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For the remaining results in this paper it is convenient, for a fixed nonnegative integer k and m > k + 1, to define
B(x) := Ak (xm , m) .
Theorem 2.2. Let p1 , . . . , p` and m be positive real numbers. If
P` :=
`
X
pi ,
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
i=1
then
`
X
(2.3)
C.E.M. Pearce and J. Pečarić
B(pj ) > ` − 1 + B (P` ) .
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j=1
Proof. We establish the result by induction, (1.2) with n = m providing a basis
(2.4)
B(p) + B(q) > 1 + B(p + q)
for p, q > 0 and p + q ≤ 1
for ` = 2. Suppose (2.3) to be true for some ` ≥ 2. Then by the inductive
hypothesis
`+1
X
j=1
B(pj ) =
`
X
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B(pj ) + B(p`+1 )
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j=1
> ` − 1 + B(P` ) + B(p`+1 ).
J. Ineq. Pure and Appl. Math. 6(1) Art. 24, 2005
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Now applying (2.4) yields
`+1
X
B(pj ) > ` − 1 + 1 + B(P` + p`+1 )
j=1
(2.5)
= ` + B(P`+1 )
as desired.
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
C.E.M. Pearce and J. Pečarić
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3.
Concavity of B
Inequality (2.3) is of the form
n
X
f (pj ) > (n − 1)f (0) + f
j=1
n
X
!
pi ,
j=1
that is, the Petrović inequality for a concave function f . A natural question is
whether B satisfies the corresponding Jensen inequality
!
n
n
1X
1X
(3.1)
B
pj ≥
B(pj )
n j=1
n j=1
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
C.E.M. Pearce and J. Pečarić
Pn
for positive p1 , p2 , . . . , pn satisfying j=1 pj ≤ 1 and indeed whether B is concave. We now address these questions. It is convenient to first deal separately
with the case n = 2.
Theorem 3.1. Suppose p, q are positive and distinct with p + q ≤ 1. Then
p+q
1
(3.2)
B
> [B(p) + B(q)] .
2
2
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Proof. Let u ∈ [0, 1). For p ∈ [0, 1 − u] we define
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G(p) = B(p) + B(1 − u − p).
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By an argument of Alzer [1] we have
(3.3)
G0 (p) =
m
k
(m − k)mpm−1 (1 − pm )m−1
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m
p
1 − pm
k
[g(p) − 1],
J. Ineq. Pure and Appl. Math. 6(1) Art. 24, 2005
http://jipam.vu.edu.au
where
(3.4)
g(p) =
1−u−p
1 − pm
m−1 m−1
1 − (1 − u − p)m
p
k k
(1 − u − p)m
1 − pm
×
1 − (1 − u − p)m
pm
is a strictly decreasing function.
It was shown in [1] that there exists p0 ∈ (0, 1 − u) such that G(p) is strictly
increasing on [0, p0 ] and strictly decreasing on [p0 , 1 − u], so that
G(p) < G(p0 )
for
p ∈ [0, 1 − u], p 6= p0 .
On the other hand, we have by (3.4) that g((1 − u)/2) = 1 and so from (3.3)
G0 ((1 − u)/2) = 0. Hence p0 = (1 − u)/2 and therefore
1−u
G(p) < G
for p 6= (1 − u)/2.
2
Set u = 1 − (p + q). Since p 6= q, we must have p 6= (1 − u)/2. Therefore
p+q
G(p) < G
,
2
which is simply (3.2).
Corollary 3.2. The map B is concave on (0, 1).
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
C.E.M. Pearce and J. Pečarić
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Proof. Theorem 3.1 gives that B is Jensen concave, so that −B is Jensen–
convex. Since B is continuous, we have by a classical result [3, Chapter 3]
that −B must also be convex and so B is concave.
The following result funishes additional information about strictness.
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Theorem 3.3. Let p1 , . . . , pn , be positive numbers with nj=1 pj ≤ 1. Then
(3.1) applies. If not all the pj are equal, then the inequality is strict.
Proof. The result is trivial with equality if the pj all share a common value, so
we assume at least two different values.
We proceed by induction, Theorem 3.1 providing a basis for nP= 2. For the
inductive step, suppose that (3.1) holds for some n ≥ 2 and that n+1
j=1 pj ≤ 1.
Without loss of generality we may assume that pn+1 is the greatest of the values
pj . Since not all the values pj are equal, we therefore have
pn+1
n
1X
>
pj .
n j=1
This rearranges to give
"
#
n
n+1
1
n−1X
1X
pj <
pn+1 +
pj .
n j=1
n
n + 1 j=1
Both sides of this inequality take values in (0, 1).
Also we have
" n
(
)#
n+1
n+1
1 X
1 1X
1
n−1X
pj =
pj +
pn+1 +
pj
.
n + 1 j=1
2 n j=1
n
n + 1 j=1
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
C.E.M. Pearce and J. Pečarić
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Hence applying (3.2) provides
n+1
1 X
pj
n + 1 j=1
B
!
"
1
>
B
2
n
1X
pj
n j=1
!
+B
1
n
(
n+1
n−1X
pn+1 +
pj
n + 1 j=1
)!#
.
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
By the inductive hypothesis
n
B
1X
pj
n j=1
!
n
1X
≥
B(pj )
n j=1
C.E.M. Pearce and J. Pečarić
Title Page
and
1
n
B
(
pn+1 +
n+1
n−1X
n+1
Contents
)!
"
1
≥
B(pn+1 ) + (n − 1)B
n
n+1
1 X
pj
n + 1 j=1
!#
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Hence
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n+1
B
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pj
j=1
1 X
pj
n + 1 j=1
!
" n+1
1 X
>
B(pj ) + (n − 1)B
2n j=1
n+1
1 X
pj
n + 1 j=1
!#
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.
J. Ineq. Pure and Appl. Math. 6(1) Art. 24, 2005
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Rearrangement of this inequality yields
!
n+1
n+1
1 X
1 X
B
pj >
B(pj ),
n + 1 j=1
n + 1 j=1
the desired result.
Remark 1. Taken together, relations (2.5) and (3.1) give
!
!
n
n
n
X
X
1X
(3.5)
n−1+B
pj <
B(pj ) ≤ nB
pj ,
n j=1
j=1
j=1
the second inequality being strict unless all the values pj are equal. If
1, this simplifies to
(3.6)
n−1<
n
X
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
Pn
j=1
pj =
C.E.M. Pearce and J. Pečarić
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−1
B(pj ) ≤ nB(n ),
Contents
j=1
since B(1) = 0.
For k = 0, (3.5) and (3.6) become (for m > 1) respectively
!m !m
!m !m
n
n
n
X
X
X
1
m
n−1+ 1−
pj
<
(1 − pm
pj
j ) ≤ n 1−
n j=1
j=1
j=1
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and
n−1<
n
X
j=1
m
−m m
(1 − pm
) .
j ) ≤ n(1 − n
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References
[1] H. ALZER, On an inequality of J.L. Brenner, J. Math. Anal. Appl., 183
(1994), 547–550.
[2] J.L. BRENNER, Analytical inequalities with applications to special functions, J. Math. Anal. Appl., 106 (1985), 427–442.
[3] G.H. HARDY, J. E. LITTLEWOOD AND G. PÓLYA, Inequalities, Cambridge University Press, Cambridge (1934).
[4] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York
(1992).
On Some Polynomial–Like
Inequalities of Brenner and
Alzer
C.E.M. Pearce and J. Pečarić
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