Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON THE HÖLDER INEQUALITY
J. PEČARIĆ AND V. ŠIMIĆ
Faculty of Textile Technology
University of Zagreb
Prilaz baruna Filipovića 30
10000 Zagreb, Croatia
volume 7, issue 5, article 176,
2006.
Received 22 November, 2006;
accepted 30 November, 2006.
Communicated by: W.S. Cheung
EMail: pecaric@hazu.hr
EMail: vidasim@ttf.hr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
301-06
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Abstract
In the present paper the authors present some new results concerning the
Hölder inequality.
2000 Mathematics Subject Classification: 26D15.
Key words: Hölder inequality, Young’s inequality, Arithmetic-geometric inequality.
A Note on the Hölder Inequality
J. Pečarić and V. Šimić
In the following, (Ω, F) is a measure space and µ is a positive measure on
Ω. Let f, g : Ω → [0, ∞) be two measurable functions. For p, q ≥ 1 such that
1
+ 1q = 1, the classical Hölder’s integral inequality is the following one ([2],
p
[3])
1q
p1 Z
q
f (x)dµ(x)
g (x)dµ(x) .
Z
Z
Ω
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p
f (x)g(x)dµ(x) ≤
(1)
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Ω
Ω
Inequality (1) may be written equivalently as
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(2)
kf gk1 ≤ kf kp kgkq ,
where
Z
kf kp =
Ω
p1
f (x)dµ(x)
p
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J. Ineq. Pure and Appl. Math. 7(5) Art. 176, 2006
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The classical proof of (1) is based on Young’s inequality
uv ≤
(3)
up v q
+ ,
p
q
where u, v ≥ 0 and p, q ≥ 1 such that p1 + 1q = 1.
Moveover, recently the following result about (3) were obtained in ([1]):
Lemma 1. Let u, v ≥ 0 and p, q ≥ 1 such that
1
p
+
1
q
= 1. Then for p ≥ 2
A Note on the Hölder Inequality
1
P (u, v) ≤ u2−p (v − up−1 )2 ,
2
(4)
where
P (u, v) =
J. Pečarić and V. Šimić
up v q
+
− uv.
p
q
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If p ∈ (1, 2], then the reverse inequality in (4) is valid. For p = 2 we have the
identity in (4).
First, we shall give a new proof of Lemma 1.
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Proof. Inequality (4) is equivalent to the following
1 2−p 2
1 1
vq
u v +
−
up −
≥ 0,
2
2 p
q
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i.e.
(5)
u2−p
q
q 2 q(p − 2) 2(p−1)
v +
u
− v q up−2
2
2p
J. Ineq. Pure and Appl. Math. 7(5) Art. 176, 2006
≥ 0.
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Let us denote by Q(u, v) the left-hand side of (5). Observe that
q q(p − 2)
+
= 1.
2
2p
Suppose that p ≥ 2, that is q ≤ 2. Using the known arithmetic-geometric
inequality ([2], [3]) we obtain
q(p−2)
q
q 2 q(p − 2) 2(p−1)
v +
u
≥ (v 2 ) 2 (u2(p−1) ) 2p ≡ v q up−2 .
2
2p
Thus Q(u, v) ≥ 0 and (5) is valid. For p ∈ (1, 2] applying the reverse
arithmetic-geometric inequality we have the reverse inequality in (5).
We will prove the next theorem.
Theorem 2. Suppose that p1 + 1q = 1 for 1 < q ≤ 2 ≤ p < ∞. Then the
following inequalities are valid
2 q
2−q p
q−1
g
f kgkq − g kf kp 1
1
(6)
≤ kf kp kgkq − kf gk1
q
2
kf kp kgkqp
2 p
2−p q
p−1
f
gkf kp − f kgkq 1
1
≤
.
p
q
2
kf kp kgkq
A Note on the Hölder Inequality
J. Pečarić and V. Šimić
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Proof. If we set in (4)
(7)
f (x)
u=
,
kf kp
g(x)
v=
,
kgkq
J. Ineq. Pure and Appl. Math. 7(5) Art. 176, 2006
http://jipam.vu.edu.au
we obtain
1 f p (x)
f (x) g(x) 1 g q (x)
1 f 2−p (x)
−
+
≤
p kf kpp
kf kp kgkq q kgkqq
2 kf k2−p
p
Integrating the last inequality, we obtain
1
kf gk1
2−p
1−
≤
f
2−p kf kp kgkq
2kf kp g(x) f p−1 (x)
−
kgkq
kf kp−1
p
2
.
! f
g
2
−
1,
p
kgkq kf k q
p−1
p
A Note on the Hölder Inequality
J. Pečarić and V. Šimić
i.e.,
kf kp kgkq − kf gk1 ≤
1
2
2 p
2−p q
p−1
f
gkf kp − f kgkq 1
p
q
,
kf kp kgkq
which proves the right-hand side of (6).
For the left-hand side of (6) we use the reverse of the inequality in (4). After
the substitutions u → v, v → u, p → q and q → p we have
1
P (u, v) ≥ v 2−q (u − v q−1 )2 .
2
For u and v from (7) we can similarly obtain the first inequality in (6).
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J. Ineq. Pure and Appl. Math. 7(5) Art. 176, 2006
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References
[1] O. DOŠLÝ AND Á. ELBERT, Integral charcterization of the principal solution of the half-linear second order differential equations, Studia Scientiarum Mathematicarum Hungarica, 36 (2000), 455–469.
[2] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities , 2nd ed.,
Cambridge Univ. Press, Cambridge, 1959.
[3] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht 1993.
A Note on the Hölder Inequality
J. Pečarić and V. Šimić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 176, 2006
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