Journal of Inequalities in Pure and
Applied Mathematics
HADAMARD-TYPE INEQUALITIES FOR QUASICONVEX FUNCTIONS
volume 4, issue 1, article 13,
2003.
N. HADJISAVVAS
Department of Product and Systems Design Engineering
University of the Aegean
84100 Hermoupolis, Syros
GREECE.
EMail: nhad@aegean.gr
URL: http://www.syros.aegean.gr/users/nhad/
Received 3 December, 2002;
accepted 19 December, 2002.
Communicated by: A.M. Rubinov
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c 2000 Victoria University
ISSN (electronic): 1443-5756
133-02
Quit
Abstract
Recently Hadamard-type inequalities for nonnegative, evenly quasiconvex functions which attain their minimum have been established. We show that these
inequalities remain valid for the larger class containing all nonnegative quasiconvex functions, and show equality of the corresponding Hadamard constants
in case of a symmetric domain.
2000 Mathematics Subject Classification: 26D15, 26B25, 39B62.
Key words: Quasiconvex function, Hadamard inequality.
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Inequality for Quasiconvex Functions . . . . . . . . . . . . . . . . . . . . 5
3
Inequality for Quasiconvex Functions such that f (0) = 0. . . . 11
References
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
1.
Introduction
The well-known Hadamard inequality for convex functions has been recently
generalized to include other types of functions. For instance, Pearce and Rubinov [2], generalized an earlier result of Dragomir and Pearce [1] by showing that
for any nonnegative quasiconvex function defined on [0, 1] and any u ∈ [0, 1],
the following inequality holds:
Z 1
1
f (u) ≤
f (x) dx.
min (u, 1 − u) 0
In a subsequent paper, Rubinov and Dutta [3] extended the result to the ndimensional space, by imposing the restriction that the nonnegative function f
attains its minimum and is not just quasiconvex, but evenly quasiconvex. The
purpose of this note is to establish the inequality without these restrictions, and
to obtain a simpler expression of the “Hadamard constant” which appears multiplied to the integral. To be precise, given a convex subset X of Rn , a Borel
measure µ on X and an element u ∈ X, we show that any nonnegative
quaR
siconvex function satisfies an inequality of the form Rf (u) ≤ γ X f dµ where
γ is a constant. An analogous inequality f (u) ≤ γ∗ X f dµ is obtained for all
quasiconvex nonnegative functions for which f (0) = 0 (under the assumption
that 0 ∈ X). We obtain simple expressions for the constants γ and γ∗ and show
that they are equal, under a symmetry assumption.
In what follows, X is a convex, Borel subset of Rn , µ is a finite Borel measure on X, and λ is the Lebesgue measure. As usual, µ λ means that µ is
absolutely continuous with respect to λ. The open (closed) ball with center u
and radius r will be denoted by B (u, r) (B (u, r)). We denote by S the sphere
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
{x ∈ Rn : kxk = 1} and set, for each v ∈ S, u ∈ R,
(1.1)
Xv,u = {x ∈ X : hv, x − ui > 0} .
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
2.
Inequality for Quasiconvex Functions
The following proposition shows that the Hadamard-type inequality for nonnegative evenly quasiconvex functions that attain their minimum, established in
[3], is true for all nonnegative quasiconvex, Borel measurable functions.
Proposition 2.1. Let f : X → R∪ {+∞} be a Borel measurable, nonnegative
quasiconvex function. Then for every u ∈ X, the following inequality holds:
Z
(2.1)
inf µ (Xv,u ) f (u) ≤
f dµ.
v∈S
X
Proof. Let L = {x ∈ X : f (x) < f (u)}. Then L is convex and u does not belong to the relative interior of L. We can thus separate u and L by a hyperplane,
i.e., there exists v ∈ S such that ∀x ∈ L, hx, vi ≤ hu, vi. Hence, for every
x ∈ Xv,u , f (x) ≥ f (u). Consequently,
Z
Z
µ (Xv,u ) f (u) ≤
f dµ ≤
f dµ
Xv,u
X
from which follows relation (2.1).
Note that if µ = λ, then we do not have to assume f to be Borel measurable.
Indeed, any convex subset of Rn is Lebesgue measurable since it can be written
as the union of its interior and a subset of its boundary; the latter is a Lebesgue
null set, thus is Lebesgue measurable. Consequently, every quasiconvex function is Lebesgue measurable since by definition its level sets are convex.
It is possible that inf v∈S µ (Xv,u ) = 0. In this case relation (2.1) does not say
much. We can avoid this if u ∈ int X and µ does not vanish on sets of nonzero
Lebesgue measure:
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
Proposition 2.2. Assumptions as in Proposition 2.1.
(i) If u ∈ int X and λ µ, then inf v∈S µ (Xv,u ) > 0.
(ii) If u ∈
/ int X and µ λ, then inf v∈S µ (Xv,u ) = 0.
Proof.
(i) Let ε > 0 be such that B (u, ε) ⊆ X. For each v ∈ S and x ∈ B u + 2ε v, 2ε ,
the triangle inequality yields
kx − uk ≤ x − u + 2ε v + 2ε v < ε
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
hence x ∈ X. Also,
hv, x − ui = v, x − u + 2ε v + v, 2ε v ≥ − kvk x − u + 2ε v + 2ε > 0.
Consequently, x ∈ Xv,u , i.e., B u + 2ε v, 2ε ⊆ Xv,u . Hence,
inf λ (Xv,u ) ≥ λ B u + 2ε v, 2ε = λ B 0, 2ε > 0.
v∈S
By absolute continuity, inf v∈S µ (Xv,u ) > 0.
Title Page
Contents
JJ
J
II
I
Go Back
Close
(ii) Since u ∈
/ int X, we can separate u and X by a hyperplane. It follows
that for some v ∈ S, the set Xv,u is a subset of this hyperplane, hence
λ (Xv,u ) = 0 which entails that µ (Xv,u ) = 0.
Quit
Page 6 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
Let us set
γ=
(2.2)
1
,
inf v∈S µ (Xv,u )
1
0
= +∞. Then we can write (2.1) in the form
Z
f (u) ≤ γ
f dµ.
where we make the convention
(2.3)
X
The following Lemma will be useful for obtaining alternative expressions of
“Hadamard constants” such as γ and showing their sharpness. In particular, it
shows that Xv,u could have been defined (see relation (1.1)) by using ≥ instead
of >. Let
(2.4)
X v,u = {x ∈ X : hv, x − ui ≥ 0}
be the closure of Xv,u in X.
Lemma 2.3. If µ λ, then
(i) µ (Xv,u ) = µ X v,u ;
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
(ii) The function v → µ (Xv,u ) is continuous on S.
Proof.
(i) We know thatλ ({x ∈ Rn : hv, x − ui = 0}) = 0; consequently,
λ X v,u \Xv,u = 0 and this entails that µ X v,u \Xv,u = 0.
Quit
Page 7 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
(ii) Suppose that (vn ) is a sequence in S, converging tov. Let ε > 0 be given.
Choose r > 0 large enough so that µ X\B (u, r) < ε/2. Let us show
that
lim λ Xvn ,u ∩ B (u, r) = λ Xv,u ∩ B (u, r) .
n→∞
For this it is sufficient to show that limn→∞ λ (Xn ) = 0 where Xn is the
symmetric difference ((Xv,u \Xvn ,u ) ∪ (Xvn ,u \Xv,u ))∩B (u, r). If x ∈ Xn
then kx − uk ≤ r and
(2.5)
hv, x − ui > 0 ≥ hvn , x − ui
N. Hadjisavvas
or
(2.6)
Hadamard-type Inequalities for
Quasiconvex Functions
hvn , x − ui > 0 ≥ hv, x − ui .
If, say, (2.6) is true, then hv, x − ui ≤ 0 < hvn − v, x − ui + hv, x − ui ≤
kvn − vk r + hv, x − ui thus |hv, x − ui| ≤ kvn − vk r. The same can be
deduced if (2.5) is true. Thus the projection of Xn on v can be arbitrarily
small; since Xn is contained in B (u, r) this means that limn→∞ λ (Xn ) =
0 as claimed.
By absolute continuity, limn→∞ µ Xvn ,u ∩ B (u, r) = µ Xv,u ∩ B (u, r) .
Since
|µ (Xvn ,u ) − µ (Xv,u )| ≤ µ Xvn ,u ∩ B (u, r) − µ Xv,u ∩ B (u, r)
+ µ Xvn ,u \B (u, r) + µ Xv,u \B (u, r)
≤ µ Xvn ,u ∩ B (u, r) − µ Xv,u ∩ B (u, r) + ε,
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
it follows that limn→∞ |µ (Xvn ,u ) − µ (Xv,u )| ≤ ε. This is true for all
ε > 0, hence limn→∞ µ (Xvn ,u ) = µ (Xv,u ).
We now obtain an alternative expression for the “Hadamard constant” γ,
analogous to the one in [3]. For u ∈ X define
n
A+
u = {(v, x0 ) ∈ R × X : hv, u − x0 i ≥ 1} .
Hadamard-type Inequalities for
Quasiconvex Functions
Further, given v ∈ Rn and x0 ∈ X set1
+
Xv,x
= {x ∈ X : hv, x − x0 i > 1} .
0
Proposition 2.4. The following equality holds for every u ∈ int X:
γ=
+
inf (v,x0 )∈A+u µ Xv,x
0
.
0
Proof. For every (v, x0 ) ∈ A+
u , we set v = v/ kvk. For each x ∈ Xv 0 ,u ,
hv, x − ui > 0 holds. Besides, (v, x0 ) ∈ A+
u implies that hv, u − x0 i ≥ 1.
+
Hence, hv, x − x0 i = hv, x − ui + hv, u − x0 i > 1 thus x ∈ Xv,x
. It follows
0
+
that Xv0 ,u ⊆ Xv,x0 ; consequently,
(2.7)
1
inf
(v,x0 )∈A+
u
µ
Title Page
Contents
1
+
Xv,x
0
N. Hadjisavvas
≥ inf µ (Xv,u ) .
v∈S
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 14
There is sometimes a change in notation with respect to [3].
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
To show the reverse inequality, let v ∈ S be given. Since u ∈ int X, we may
find x0 ∈ X such that hv, u − x0 i > 0. Choose t > 0 so that for v 0 = tv one
has hv 0 , u − x0 i = 1. The following equivalences hold:
x ∈ Xv+0 ,x0 ⇔ hv 0 , x − x0 i > 1
⇔ hv 0 , x − ui + hv 0 , u − x0 i > 1
⇔ hv 0 , x − ui > 0
⇔ hv, x − ui > 0
⇔ x ∈ Xv,u .
+
Thus, for every v ∈ S there exists (v 0 , x0 ) ∈ A+
u such that Xv,u = Xv 0 ,x0 . Hence
equality holds in (2.7).
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
3.
Inequality for Quasiconvex Functions such that
f (0) = 0.
Whenever 0 ∈ X and f (0) = 0, another Hadamard-type inequality has being
obtained in [3], assuming that f is nonnegative and evenly quasiconvex. We
generalize this result to nonnegative quasiconvex functions and compare with
the previous findings. Let h : R+ → R+ be increasing with h (c) > 0 for all
c
c > 0 and λh := supc>0 h(c)
< +∞ (we follow the notation of [3]).
Proposition 3.1. Let f : X → R ∪ {+∞} be Borel measurable, nonnegative
and quasiconvex. If 0 ∈ X and f (0) = 0, then for every u ∈ X,
Z
(3.1)
inf
µ (Xv,u ) f (u) ≤ λh
h (f (x)) dµ.
v∈S,hv,ui≥0
Proof. If f (u) = 0 we have nothing to prove. Suppose that f (u) > 0. Coming
back to the proof of Proposition 2.1, we know that there exists v ∈ S such that
∀x ∈ Xv,u , f (x) ≥ f (u); hence h (f (x)) ≥ h (f (u)), from which it follows
that
Z
Z
µ (Xv,u ) h (f (u)) ≤
h (f (x)) dµ ≤
h (f (x)) dµ.
X
Note that 0 ∈
/ Xv,u because f (0) < f (u); thus, hv, ui ≥ 0. Consequently,
Z
inf
µ (Xv,u ) h (f (u)) ≤
h (f (x)) dµ.
v∈S,hv,ui≥0
N. Hadjisavvas
Title Page
X
Xv,u
Hadamard-type Inequalities for
Quasiconvex Functions
X
Finally, note that by definition of λh , f (u) ≤ λh h (f (u)) from which follows (3.1).
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
Note that relation (3.1) is only interesting if u 6= 0 since otherwise it is
trivially true. Let us define γ∗ by

1

if u 6= 0
inf v∈S,hv,ui≥0 µ (Xv,u )
(3.2)
γ∗ =

0
if u = 0.
We obtain an alternative expression for γ∗ , similar to that in [3]. Given
u ∈ X\ {0}, set Bu = {v ∈ Rn : hv, ui ≥ 1}, and for any v ∈ Rn , set Xv+ =
{x ∈ X : hv, xi > 1}.
Proposition 3.2. The following equality holds for every u ∈ X\ {0}:
inf
µ (Xv,u ) .
inf µ Xv+ =
v∈Bu
v∈S,hv,ui>0
Proof. For every v ∈ Bu we set v 0 = v/ kvk and show that Xv0 ,u ⊆ Xv+ . Indeed,
if x ∈ Xv0 ,u , then we have hv, x − ui > 0 hence hv, xi = hv, ui + hv, x − ui >
1, i.e., x ∈ Xv+ . Since hv 0 , ui > 0, it follows that
(3.3)
inf µ Xv+ ≥
inf
µ (Xv,u ) .
v∈Bu
v∈S,hv,ui>0
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
To show equality, let v ∈ S be such that hv, ui > 0. Choose t > 0 such that
t hv, ui = 1 and set v 0 = tv. For every x ∈ Xv+0 one has hv 0 , xi > 1, hence,
Close
hv 0 , x − ui = hv 0 , xi − hv 0 , ui > 0.
Page 12 of 14
It follows that hv, x − ui > 0, i.e., x ∈ Xv,u . Thus, Xv+0 ⊆ Xv,u and v 0 ∈ Bu .
This shows that in (3.3) equality holds.
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
Quit
http://jipam.vu.edu.au
Proposition 3.3. If µ λ then we also have the equalities
1
1
(if u 6= 0)
γ∗ =
=
inf v∈S,hv,ui>0 µ (Xv,u )
minv∈S,hv,ui≥0 µ X v,u
1
.
(3.4)
γ=
minv∈S µ X v,u
Proof. We first observe that, according to Lemma 2.3, µ X v,u = µ (Xv,u ).
The same lemma entails that inf v∈S,hv,ui>0 µ (Xv,u ) = inf v∈S,hv,ui≥0 µ (Xv,u )
and that this infimum is attained, since the set {v ∈ S : hv, ui ≥ 0} is compact.
In the same way, the infimum in (2.2) is attained.
Whenever µ λ, the constant γ is sharp, in the sense that given Ru ∈ X,
there exists a nonnegative quasiconvex function f such that f (u) = γ X f dµ.
Indeed, since the minimum in (3.4) is attained for some v0 ∈ S, it is sufficient
to take f to be the characteristic function of X v0 ,u (see Corollary 2 of [3]).
Analogous considerations can be made for γ∗ (see Corollary 4 of [3]).
We now show the equality of γ and γ∗ under a symmetry assumption:
Corollary 3.4. Suppose that X has 0 as center of symmetry, u ∈ int X\ {0}
and µ λ. If µ (A) = µ (−A) for every Borel A ⊆ X, then γ = γ∗ .
0
Proof. For every v ∈ S such that hv, ui < 0, set v = −v and Y = {x ∈
X : hv, x + ui > 0}. Since 0 is a center of symmetry, one can check that
Y = −Xv0 ,u .
If x ∈ Y then hv, x − ui = hv, x + ui − 2 hv, ui > 0. Thus, Y ⊆ Xv,u
and µ (Xv,u ) ≥ µ (Y ) = µ (Xv0 ,u ). It follows that the minimum in (3.4) can be
restricted to v ∈ S such that hv, ui ≥ 0. Thus, γ = γ∗ .
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au
References
[1] S.S. DRAGOMIR AND C.E.M. PEARCE, Quasi-convex functions and
Hadamard’s inequality, Bull. Austral. Math. Soc., 57 (1998), 377–385.
[2] C.E.M. PEARCE AND A.M. RUBINOV, P –functions, quasiconvex functions and Hadamard-type inequalities, J. Math. Anal. Appl., 240 (1999),
92–104.
[3] A.M. RUBINOV AND J. DUTTA, Hadamard type inequalities for quasiconvex functions in higher dimensions, J. Math. Anal. Appl., 270 (2002),
80–91.
Hadamard-type Inequalities for
Quasiconvex Functions
N. Hadjisavvas
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 14
J. Ineq. Pure and Appl. Math. 4(1) Art. 13, 2003
http://jipam.vu.edu.au