Journal of Inequalities in Pure and
Applied Mathematics
GENERALIZED INEQUALITIES FOR INDEFINITE FORMS
FATHI B. SAIDI
Mathematics Division,
University of Sharjah,
P. O. Box 27272, Sharjah,
United Arab Emirates.
EMail: fsaidi@sharjah.ac.ae
volume 4, issue 5, article 95,
2003.
Received 19 February, 2003;
accepted 31 July, 2003.
Communicated by: L. Losonczi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
016-03
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Abstract
We establish abstract inequalities that give, as particular cases, many previously established Hölder-like inequalities. In addition to unifying the proofs of
these inequalities, which, in most cases, tend to be technical and obscure, the
proofs of our inequalities are quite simple and basic. Moreover, we show that
sharper inequalities can be obtained by applying our results.
2000 Mathematics Subject Classification: Primary 26D15, 26D20
Key words: Inequalities, Holder, Indefinite forms, Reverse Inequalities, Holder’s Inequality, Generalized inequalities.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Generalized Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
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1.
Introduction
Let nP
≥ 2 be a fixed integerP
and let ai , bi ∈ R, i = 1, 2, . . . , n, be such that
n
2
2
2
a1 − i=2 ai ≥ 0 and b1 − ni=2 b2i ≥ 0, where R is the set of real numbers.
Then
! 21
! 12
n
n
n
X
X
X
2
2
2
2
(1.1)
a1 −
ai
b1 −
bi
≤ a1 b 1 −
ai b i .
i=2
i=2
i=2
This inequality was first considered by Aczél and Varga [2]. It was proved in
detail by Aczél [1], who used it to present some applications of functional equations in non-Euclidean geometry. Inequality (1.1) was generalized by Popoviciu [8] as follows. Let p > 1, p1 + 1q = 1, ai , bi ≥ 0, i = 1, 2, . . . , n, with
P
P
ap1 − ni=2 api ≥ 0 and bq1 − ni=2 bqi ≥ 0. Then
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
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(1.2)
ap1 −
n
X
! p1
api
i=2
bq1 −
n
X
! 1q
bqi
≤ a1 b 1 −
n
X
i=2
ai b i .
i=2
This is the “Hölder-like” generalization of (1.1). A simple proof of (1.2) may
be found in [10]. Also, Chapter 5 in [6] contains generalizations of (1.2).
For a fixed integer n ≥ 2 and p(6= 0) ∈ R, the authors in [5] introduced the
following definition:
(1.3)
Φp (x) :=
xp1 −
n
X
i=2
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! p1
xpi
JJ
J
, x ∈ Rp ,
J. Ineq. Pure and Appl. Math. 4(5) Art. 95, 2003
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where
(
(1.4)
Rp =
x = (x1 , . . . , xn ) : xi ≥ (>) 0,
xp1
≥ (>)
n
X
)
xpi
i=2
if p > (<)0.
There they presented inequalities for Φp from which they deduced, among other
things, the inequalities (1.1) and (1.2).
Finally, in [9] the authors introduced the following definitions, which generalize (1.3) and (1.4). Let n be a positive integer, n ≥ 2, and let M be a one-toone real-valued function whose domain is a subset of R. Then, for α ∈ R,
Rα,M = x = (x1 , x2 , . . . , xn ) : x1 > 0,
xi
∈ Domain (M ) for i = 2, . . . , n,
x1
"
#
n
X
xi
and α −
M
∈ Range (M )
x1
i=2
and, for x ∈ Rα,M ,
Fathi B. Saidi
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Φα,M (x) = x1 M
Generalized Inequalities for
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−1
α−
n
X
i=2
M
xi
x1
#
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.
There the authors obtained generalizations of inequalities (1.1) and (1.2) and of
the inequalities in [5].
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It is our aim in this paper to establish inequalities (see Theorems 2.1 and 2.2)
that give, as particular cases, all the inequalities mentioned above. In addition
to unifying the proofs of these inequalities, which, in most cases, tend to be
technical and obscure in the sense that it is not clear what really makes them
work, the proofs of our inequalities are quite simple and basic. Moreover, we
show that sharper inequalities can be obtained by applying our results.
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
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2.
Generalized Inequalities
Let Rα,M and Φα,M be as defined above and let m ≥ 2 be an integer.
Theorem 2.1. Let M1 , M2 , . . . , Mm be one-to-one real-valued functions defined in R and let M be a real-valued function defined on Domain (M1 ) ×
· · · × Domain (Mm ) and satisfying, for all (t1 , . . . , tm ),
(2.1)
M (t1 , t2 , . . . , tm ) ≤ (≥)
m
X
σk Mk (tk ) ,
k=1
where σ1 , σ2 , . . . σm are fixed real numbers. Then
Φα1 ,M1 (x1 )
Φαm ,Mm (xm )
(2.2) M
,...,
x11
xm1
m
n
X
X
xmi
x1i
≤ (≥)
σk αk −
M
,...,
x
xm1
11
i=2
k=1
for all αk ∈ R satisfying Rαk ,Mk 6= ∅ and all xk ∈ Rαk ,Mk , k = 1, . . . , m.
Proof. Using (2.1) and the definition of Φαk ,Mk (xk ), we obtain
Φα1 ,M1 (x1 )
Φαm ,Mm (xm )
M
,...,
x11
xm1
m
X
Φαk ,Mk (xk )
≤ (≥)
σk Mk
xk1
k=1
"
#
m
n
X
X
xki
Mk
=
σk αk −
xk1
i=2
k=1
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
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=
m
X
σk αk −
k=1
≤ (≥)
m
X
k=1
n X
m
X
i=2 k=1
n
X
σk αk −
σk Mk
M
i=2
xki
xk1
x1i
xmi
,...
x11
xm1
.
This ends the proof.
Theorem 2.1, besides giving a unified and much simpler proof, is more general than many previously established inequalities. Indeed, as is shown below
in the remarks following Corollary 3.2, these inequalities can be obtained as
consequences of Theorem
2.1 with appropriate choices for the Mk ’s and with
Q
M (t1 , . . . , tm ) := m
t
.
k=1 k
Moreover, since inequality (2.2) is sharper whenever M is larger (smaller),
we can obtain sharper inequalities each time we keep the same Mk ’s while modifying M so that the surface tm+1 = M (t1 , . . . tm ) in Rm+1
distinct from and
Qis
m
is between the twoPsurfaces tm+1 = P (t1 , . . . , tm ) := k=1 tk and tm+1 =
S (t1 , . . . , tm ) := m
k=1 σk Mk (tk ). In other words, each time we chose M 6=
P, S such that, for every (t1 , . . . , tm ) ∈ Domain (M1 ) × · · · × Domain (Mm ),
Generalized Inequalities for
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Fathi B. Saidi
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(2.3)
m
Y
k=1
tk ≤ (≥) M (t1 , . . . tm ) ≤ (≥)
m
X
σk Mk (tk ) .
k=1
The closer M gets to
the sharper the inequality is. Clearly, the optimum M is
PS,
m
M (t1 , . . . , tm ) := k=1 σk Mk (tk ), in which case equality is attained in (2.2).
But the idea is to choose an M that satisfies (2.3) while being simple enough
to yield a “nice inequality”. This, of course is most useful when the Mk ’s are
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not that simple. Nevertheless, any choice of M satisfying (2.3) will give a new
inequality, strange as it may look.
To further clarify the above remarks, we establish the following consequence
of Theorem 2.1, in which it is apparent that previous inequalities are particular
cases and that Theorem 2.1 does indeed lead to sharper inequalities:
Theorem 2.2. Let M1 , M2 , . . . , Mm be one-to-one real-valued functions defined in R and satisfying, for all (t1 , . . . , tm ) ∈ Domain(M1 )×· · ·×Domain(Mm ),
m
Y
(2.4)
tk ≤ (≥)
k=1
m
X
Generalized Inequalities for
Indefinite Forms
σk Mk (tk ) ,
k=1
Fathi B. Saidi
where σ1 , σ2 , . . . , σm are fixed real numbers. Let µ be any real-valued function defined on Domain (M1 ) × · · · × Domain (Mm ) and satisfying, for every
(t1 , . . . , tm ),
0 ≤ µ(t1 , . . . , tm ) ≤ 1.
Then
!
! m
m
m
n Y
m
X
Y
X
Y
xki
(2.5)
Φαk ,Mk (xk ) ≤ (≥)
σk αk
xk1 −
k=1
k=1
−
n X
i=2
k=1
i=2 k=1
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x1i
xmi
1−µ
,...,
x11
xm1
! m
Y
m
m
X
xki
xki Y
×
σk Mk
−
xk1
xk1
xk1 k=1
k=1
k=1
for all αk ∈ R satisfying Rαk ,Mk 6= ∅ and all xk ∈ Rαk ,Mk , k = 1, . . . , m.
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Proof. For simplicity of notation, let
α :=
m
X
σk αk , P (Φ) :=
k=1
S(Φ) :=
m
X
σk Mk
µ(Φ) := µ
(xk )
k ,Mk
xk1
k=1
k=1
m
Y
Φα
Φαk ,Mk (xk )
xk1
m
Y
xki
, Pi (x) :=
,
xk1
k=1
, Si (x) :=
m
X
σk Mk
k=1
Φαk ,Mk (xk )
Φα ,M (xk )
,..., k k
xk1
xk1
, µi (x) := µ
xki
xk1
,
x1i
xmi
,...,
x11
xm1
.
Generalized Inequalities for
Indefinite Forms
Let
M (t1 , . . . , tm ) :=
m
Y
tk + (1 − µ (t1 , . . . , tm ))
k=1
m
X
k=1
σk Mk (tk ) −
m
Y
Fathi B. Saidi
!
tk
.
k=1
Then M satisfies the inequalities in (2.3). Therefore we may apply Theorem
2.1 to obtain
P (Φ) + (1 − µ (Φ)) (S (Φ) − P (Φ))
n
X
≤ (≥) α −
(Pi (x) + (1 − µi (x)) (Si (x) − Pi (x))) .
i=2
P (Φ) ≤ (≥) α −
i=2
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Rearranging the terms, we get
n
X
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!
Pi (x)
−
n
X
!
(1 − µi (x)) (Si (x) − Pi (x))
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i=2
− (1 − µ(Φ)) (S(Φ) − P (Φ)) .
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Since the inequalities in (2.3) hold, we may drop the last term to obtain
!
!
n
n
X
X
P (Φ) ≤ (≥) α −
Pi (x) −
(1 − µi (x)) (Si (x) − Pi (x)) .
i=2
Multiplying both sides by
(2.5). This ends the proof.
i=2
Qm
k=1
xk1 , which is positive, we obtain the result
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
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3.
Applications
Let p1 , p2 , . . . , pm 6= 0 be real numbers satisfying
well known that
(3.1)
1
p1
+
1
p2
+ ··· +
1
pm
= 1. It is
m
Y
m
X
1 pk
tk ≤
t ,
pk k
k=1
k=1
m
for every (t1 , . . . , tm ) ∈ Rm
+ := (0, ∞) , if and only if all pi ’s are positive.
Inequality (3.1) is known as Hölder’s inequality.
Also, one has the following reverse inequality to (3.1):
(3.2)
m
Y
k=1
tk ≥
m
X
1 pk
tk ,
p
k
k=1
for every (t1 , . . . , tm ) ∈ Rm
+ , if and only if all pi ’s are negative except for exactly
one of them, [9] and [11].
Setting Mk (t) := tpk , σk = p1k , and αk = 1, k = 1, . . . , m, in Theorem 2.2,
we obtain immediately the following corollary:
Corollary 3.1. Let p1 , p2 , . . . , pm 6= 0 be real numbers satisfying p11 + p12 +· · ·+
1
= 1 and let µ be any real-valued function defined on Rm
+ and satisfying, for
pm
every (t1 , . . . , tm ),
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Fathi B. Saidi
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(3.3)
0 ≤ µ(t1 , . . . , tm ) ≤ 1.
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If all pi ’s are positive (all pi ’s are negative except for exactly one of them), then
(3.4)
m
Y
xpk1k
−
n
X
! p1
m
Y
k
xpkik
≤ (≥)
i=2
k=1
xk1 −
k=1
−
n X
i=2
n Y
m
X
!
xki
i=2 k=1
x1i
xmi
1−µ
,...,
x11
xm1
! m
pk Y
m
m
X
1 xki
xki Y
×
−
xk1
pk xk1
xk1 k=1
k=1
k=1
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
for all xk ∈ R
1,tpk
, k = 1, . . . , m.
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Dropping the last term in (3.4), we obtain Corollary 1 of [5]:
Corollary 3.2. Let p1 , p2 , . . . , pm 6= 0 be real numbers satisfying
1
= 1. The inequality
pm
(3.5)
m
Y
k=1
Φpk (xk ) ≤ (≥)
m
Y
k=1
xk1 −
n Y
m
X
1
p1
+ p12 +· · ·+
xki
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holds for all xk ∈ Rpk , k = 1, . . . , m, if and only if all pk ’s are positive (all pk ’s
are negative except for exactly one of them).
Note that inequality (3.4) is sharper than inequality (3.5). Choosing µ ≡
1, (3.4) gives (3.5). But any other choice of µ, satisfying (3.3), will give a
sharper inequality. Of course, one may choose µ ≡ 0 to obtain the sharpest
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inequality from (3.4). But, by keeping µ in (3.4), we give ourselves the freedom
of choosing µ in such a way as to make the last term in (3.4) as simple as
possible. This is a trade we have to make between the sharpness of inequality
(3.4) and its simplicity.
Finally, we note that inequalities (1.1) and (1.2) are particular cases of inequality (3.5) and, consequently, of inequality (3.4).
We conclude by noting that from Páles’s paper [7] and from Losonczi’s papers [3] and [4] it follows that inequalities (3.1) and (3.2), written in the form
m
Y
m
X
tpkk − 1
tk − 1 ≤ (≥)
, (t1 , t2 , . . . , tm ) ∈ Rm ,
pk
k=1
k=1
are equivalent to
(3.6)
Mn,1
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
Title Page
m
Y
k=1
!
xk
≤ (≥)
m
Y
Contents
Mn,pk (xk ), n ∈ N, xk ∈
Rn+ ,
k=1
k = 1, 2, . . . , m,
where xk := (xk1 , xk2 , . . . , xkn ), k = 1, 2, . . . , m, and
 Pn xpj p1


if p 6= 0,
j=1 n
Mn,p (x) := Mn,p (x1 , x2 , . . . , xn ) :=

 √
n
x1 x2 · · · xn if p = 0.
Inequality (3.6) was completely settled by Páles, [7, corollary on p. 464].
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References
[1] J. ACZÉL, Some general methods in the theory of functional equations in
one variable, New applications of functional equations, Uspehi. Mat. Nauk
(N.S.) (Russian), 69 (1956), 3–68.
[2] J. ACZÉL AND O. VARGA, Bemerkung zur Cayley-Kleinschen Massbestimmung, Publ. Mat. (Debrecen), 4 (1955), 3-15.
[3] L. LOSONCZI, Subadditive Mittelwerte, Arch. Math., 22 (1971), 168–
174.
[4] L. LOSONCZI, Inequalities for integral means, J. Math. Anal. Appl., 61
(1977), 586–606.
[5] L. LOSONCZI AND Z. PÁLES, Inequalities for indefinite norms, J. Math.
Anal. Appl., 205 (1997), 148–156.
[6] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
Generalized Inequalities for
Indefinite Forms
Fathi B. Saidi
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[7] Z. PÁLES, On Hölder-type inequalities, J. Math. Anal. Appl., 95 (1983),
457–466.
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[8] T. POPOVICIU, On an inequality, Gaz. Mat. Fiz. A, 64 (1959), 451–461
(in Romanian).
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[9] F. SAIDI AND R. YOUNIS, Generalized Hölder-like inequalities, Rocky
Mountain J. Math., 29 (1999), 1491–1503.
J. Ineq. Pure and Appl. Math. 4(5) Art. 95, 2003
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[10] P.M. VASIĆ AND J.E. PEČARIĆ, On the Hölder and related inequalities,
Mathematica (Cluj), 25 (1982), 95–103.
[11] XIE-HUA SUN, On generalized Hölder inequalities, Soochow J. Math.,
23 (1997), 241–252.
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Indefinite Forms
Fathi B. Saidi
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