Journal of Inequalities in Pure and
Applied Mathematics
CONVEX FUNCTIONS AND INEQUALITIES FOR INTEGRALS
ZHENGLU JIANG, XIAOYONG FU AND HONGJIONG TIAN
Department of Mathematics
Zhongshan University
Guangzhou 510275, China
EMail: mcsjzl@mail.sysu.edu.cn
EMail: mcsfxy@mail.sysu.edu.cn
Department of Mathematics
Shanghai Teachers’ University
Shanghai 200234, China
EMail: hongjiongtian@263.net
volume 7, issue 5, article 184,
2006.
Received 04 June, 2005;
accepted 01 December, 2006.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper we present inequalities for integrals of functions that are the composition of nonnegative convex functions on an open convex set of a vector
space Rm and vector-valued functions in a weakly compact subset of a Banach
vector space generated by m Lpµ -spaces for 1 ≤ p < +∞ and inequalities when
these vector-valued functions are in a weakly* compact subset of a Banach
vector space generated by m L∞
µ -spaces.
2000 Mathematics Subject Classification: Primary 26D15.
Key words: Convex functions; Measure spaces; Lebesgue integrals; Banach
spaces.
This work was supported by grants of NSFC 10271121 and joint grants of NSFC
10511120278/10611120371 and RFBR 04-02-39026. This work was also sponsored
by SRF for ROCS, SEM.
We would like to thank the referee of this paper for his/her valuable comments on
this work.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Inequalities for Weakly Convergent Sequences . . . . . . . . . . . . 7
3
Inequalities for Weakly* Convergent Sequences . . . . . . . . . . . 11
References
Convex Functions and
Inequalities for Integrals
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Hongjiong Tian
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1.
Introduction
When studying extremum problems and integral estimates in many areas of applied mathematics, we may require the convexity of functions and the weak
compactness of sets. Many properties of convex functions and weakly compact
sets can be found in the literature (e.g., see [2], [3], [4], [5] and [7]). In some research fields such as the existence of solutions of differential equations (e.g., see
[1] and [6]), we usually enconter some problems on the estimates of integrals
of functions that are the composition of convex functions on an open convex set
of a vector space and vector-valued functions in a weakly (or weakly*) compact
subset in a Banach space. The estimates of integrals of this kind of composite function is interesting and important in many application areas. Inequalities
for integrals of composite functions are necessary, therefore, for solving many
problems in applied mathematics.
Let us first introduce some notations which will be used throughout this paper. R denotes the real number system, Rn is the usual vector space of real
n-tuples x = (x1 , x2 , . . . , xn ), µ is a nonnegative Lebesgue measure of Rn ,
Lpµ (Rn ) represents a Banach space where each measurable function u(x) has
the following norm
Z
(1.1)
kukp =
p1
|u(x)| dµ
p
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for any p ∈ [1, +∞), (Lpµ (Rn ))m denotes a Banach vector space where each
n
measurable vector-valued function has m components in Lpµ (Rn ), L∞
µ (R ) represents a Banach space where each measurable function u(x) has the following
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norm
(1.2)
kuk∞ = ess sup |u(x)| (or say kuk∞ =
x∈Rn
inf
sup |u(x)|),
E⊆Rn x∈E
µ(E C )=0
n m
where E C represents the complement set of E in Rn , and (L∞
µ (R )) denotes
a Banach vector space where each measurable vector-valued function has m
n
components in L∞
µ (R ).
Below, we give the definition of weak convergence of a sequence in Lpµ (Rn ),
where p ∈ [1, +∞). Assume that q = p/(p−1) as p ∈ (1, +∞) and that q = ∞
p
n
as p = 1. If u ∈ Lpµ (Rn ) and it is assumed that a sequence {ui }+∞
i=1 in Lµ (R )
satisfies
Z
Z
(1.3)
lim
ui vdµ =
uvdµ
i→+∞
Rn
Rn
for all v ∈ Lqµ (Rn ), then the sequence {ui }+∞
i=1 is said to be weakly convergent
p
n
in Lµ (R ) to u as i → +∞. Similarly, we introduce the definition of weak*
n
∞
n
convergence of a sequence in L∞
µ (R ). If u ∈ Lµ (R ) and it is assumed that
n
1
n
∞
a sequence {ui }+∞
i=1 in Lµ (R ) satisfies the equality (1.3) for all v ∈ Lµ (R ),
+∞
∞
n
then the sequence {ui }i=1 is said to be weakly* convergent in Lµ (R ) to u
as i → +∞. Then we define the weak (or weak*) convergence of a sequence
+∞
n m
in (Lpµ (Rn ))m (or (L∞
µ (R )) ). If 1 ≤ p < +∞ and {uji }i=1 is weakly convergent in Lpµ (Rn ) to ûj for all j = 1, 2, . . . , m as i → +∞, then a sequence
{ui = (u1i , u2i , . . . , umi )}+∞
i=1 is called weakly convergent in a Banach vector
p
n m
space (Lµ (R )) to û = (û1 , û2 , . . . , ûm ) as i → +∞. Similarly, if {uji }+∞
i=1 is
n
(R
)
to
û
for
all
j
=
1,
2,
.
.
.
,
m
as
i
→
+∞,
then
weakly* convergent in L∞
j
µ
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a sequence {ui = (u1i , u2i , . . . , umi )}+∞
i=1 is said to be weakly* convergent in a
∞
n m
Banach vector space (Lµ (R )) to û = (û1 , û2 , . . . , ûm ) as i → +∞.
We now recall the definition of a convex function. If f (x) is a function
with its values being real numbers or ±∞ and its domain is a subset S of an
m dimensional vector space Rm such that {(x, y)|x ∈ S, y ∈ R, y ≥ f (x)}
is convex as a subset of an m + 1 dimensional vector space Rm+1 , then f (x)
is called a convex function on S. It is known that f (x) is convex from S to
(−∞, +∞] if and only if
(1.4) f (λ1 x1 + λ2 x2 + · · · + λk xk ) ≤ λ1 f (x1 ) + λ2 f (x2 ) + · · · + λk f (xk )
whenever S is a convex subset of Rm , xi ∈ S (i = 1, 2, . . . ) λ1 ≥ 0, λ2 ≥ 0,
. . . , λk ≥ 0, λ1 + λ2 + · · · + λk = 1. This is called Jensen’s inequality when
S = Rm .
There are inequalities for integrals of functions that are the composition of
convex functions on a vector space Rm and vector-valued functions in a weakly*
n m
compact subset of (L∞
µ (R )) (see [6] or Theorem 3.2 in Section 3). If F (x, y)
is a special nonnegative convex function defined by F (x, y) = (x−y) log( xy ) for
(x, y) ∈ (0, +∞) × (0, ∞) and {(ai , bi )}+∞
i=1 is a nonnegative sequence weakly
1
n 2
convergent in (Lµ (R )) to (a, b), then similar inequalities for integrals of the
composite functions can also be obtained (see [1]), as follows:
Z
Z
lim
F (ai , bi )dµ ≥
F (a, b)dµ.
i→+∞
Rn
Rn
Below, we extend the two results mentioned above to a more general case. More
precisely, this paper aims to show inequalities for integrals of functions that are
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the composition of nonnegative convex functions on an open convex set of a
vector space Rm and vector-valued functions in a weakly compact subset of a
Banach vector space generated by m Lpµ -spaces for 1 ≤ p < +∞. We also show
inequalities for integrals of functions when these vector-valued functions are in
a weakly* compact subset of a Banach vector space generated by m L∞
µ -spaces.
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2.
Inequalities for Weakly Convergent Sequences
Some basic concepts have been introduced in the previous section. In this section we show inequalities for integrals of functions which are the composition of
nonnegative convex functions on an open convex set of a vector space Rm and
vector-valued functions in a weakly compact subset of a Banach vector space
generated by m Lpµ -spaces for any given p ∈ [1, +∞). That is the following
p
n m
Theorem 2.1. Suppose that a sequence {ui }+∞
i=1 weakly converges in (Lµ (R ))
to u as i → +∞, where p ∈ [1, +∞) and m and n are two positive integers.
Assume that all the values of u and ui (i = 1, 2, 3, . . . ) belong to an open convex
set K in Rm and that f (x) is a nonnegative convex function from K to R. Then
Z
Z
(2.1)
lim
f (ui )dµ ≥
f (u)dµ
i→+∞
Ω
Ω
for any measurable set Ω ⊆ Rn .
In order to prove Theorem 2.1, let us first recall the following lemma:
Lemma 2.2. Assume un → u weakly in aP
normed linear space.
PnThen there
n
exists, for any ε > 0, a convex combination
k=1 λk uk (λk ≥ 0,
k=1 λk = 1)
P
of {uk : k = 1, 2, . . . } such that ku − nk=1 λk uk k ≤ ε where kvk is a norm of
v in the space.
This is called Mazur’s lemma. Its proof can be found in [5] and [7]. Using
this lemma, we can give a proof of Theorem 2.1.
Proof of Theorem 2.1. Let ΩR be a bounded set defined
by ΩR = Ω ∩ {w :
R
n
|w| < R, w ∈ R } for all R > 0. Put αi = ΩR f (ui )dµ (i = 1, 2, . . . )
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and α = lim
i→+∞
R
ΩR
f (ui )dµ for all the bounded sets ΩR . Then there exists a
subsequence of {αi }+∞
i=1 such that this subsequence, denoted without loss of
+∞
generality by {αi }i=1
, converges to α as i → +∞.
Take Kl = {x : x ∈ K, |x| ≤ l, ρ(∂K, x) ≥ 1/l} and Dl (u) = {ω :
ω ∈ Rn , u(ω) ∈ Kl } for any fixed positive integer l, where ρ(∂K, x) is defined
as a distance between the point x and the boundary ∂K of K. Then all the
sets Kl are bounded, closed and convex subsets of K such that lim Kl = K
l→+∞
and Kl ⊂ Kl+1 for any positive integer l. It can also be easily proven that
lim Dl (u) = Rn and that Dl (u) ⊆ Dl+1 (u) for any positive integer l. Since
Convex Functions and
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l→+∞
f (x) is a convex function defined on an open convex set K, f (x) is continuous
in K (see [3]). Thus f (x) is uniformly continuous in Kl+1 , that is, for any given
positive number ε, there exists a positive number δ such that
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(2.2)
|f (x) − f (y)| < ε/µ(ΩR )
as |x − y| < δ for any x and y in Kl+1 .
Since ui weakly converges in (Lpµ (Rn ))m to u, using Lemma 2.2, one know
PN (j)
that for any natural number j, thereexists a convex combination
k=j λk uk of
PN (j) (j) {uk : k = j, j + 1, . . . } such that u − k=j λk uk p ≤ 1j , where N (j) is a
natural number which depends on j and {uk : k = j, j +1, . . . }, kvkp represents
PN (j) (j)
PN (j) (j)
(j)
a norm of v in (Lpµ (Rn ))m , λk ≥ 0 and k=j λk = 1. Put vj = k=j λk uk .
Then, as j tends to infinity, vj converges in (Lpµ (Rn ))m to u. Thus there exists
a subsequence of {vj }+∞
j=1 such that this subsequence, denoted without loss of
n
generality by {vj }+∞
,
j=1 converges almost everywhere in R to u as j goes to
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infinity. In particular, for any given R > 0, vj converges almost everywhere in
ΩR to u as j goes to infinity. By Egorov’s theorem, it is known that for any given
positive number σ, there exists a measurable set ER in ΩR with µ(ER ) < σ such
that vj converges uniformly in ΩR \ER to u. Therefore for the above number δ,
there exists a natural number N such that
|u(ω) − vj (ω)| < δ
sup
(2.3)
ω∈(ΩR \ER )∩Dl (u)
for all j > N. Since all the values of u in Dl (u) are in the closed set Kl and
Kl ⊂ Kl+1 , δ can be chosen to be sufficiently small such that all the values of
vj in (ΩR \ER ) ∩ Dl (u) fall in Kl+1 and (2.2) and (2.3) still hold for this choice
of δ. Combining (2.2) and (2.3), we know that for any given positive number ε,
there exists a natural number N such that
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{ui }+∞
i=1
in (ΩR \ER ) ∩ Dl (u) for all j > N. Since all the values of
K and f (x) is convex and nonnegative, integrating (2.4) gives
N (j)
Z
f (u)dµ ≤
(2.5)
ΩR ∩Dl (u)
X
(j)
λk
k=j
f (uk )dµ + σ max f (x) + ε.
x∈Kl
ΩR ∩Dl (u)
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N (j)
f (u)dµ ≤
(2.6)
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(2.5) can be equivalently written as
Z
and u are in
Z
ΩR
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f (u) < f (vj ) + ε/µ(ΩR )
(2.4)
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X
k=j
(j)
λk αk
+ σ max f (x) + ε.
x∈Kl
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Notice that αk → α as k → +∞. By first letting j → +∞ and then ε → 0,
(2.6) gives
Z
(2.7)
f (u)dµ ≤ α + σ max f (x).
x∈Kl
ΩR ∩Dl (u)
The fact that lim Dl (u) = Rn shows that the limit of ΩR ∩ Dl (u) in (2.7) is
l→+∞
ΩR . By first letting σ → 0 and then l → +∞, (2.7) reads
Z
Z
(2.8)
f (u)dµ ≤ α ≤ lim
f (ui )dµ,
ΩR
i→+∞
Ω
where the last inequality is obtained using the nonnegativity of f (x). Finally,
by letting R → +∞ and using the Lebesgue dominated convergence theorem,
(2.8) leads to (2.1). This completes the proof.
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3.
Inequalities for Weakly* Convergent Sequences
In the previous section we have given inequalities for integrals of composite
functions for weakly convergent sequences in (Lpµ (Rn ))m for 1 ≤ p < +∞.
In this section we present a similar result for weakly* convergent sequences in
n m
(L∞
µ (R )) .
Using the process of the proof in Theorem 2.1, we can prove the following
theorem.
∞
n m
Theorem 3.1. Assume that a sequence {ui }+∞
i=1 weakly* converges in (Lµ (R ))
to u as i → +∞, where m and n are two positive integers. Assume that all the
values of u and ui (i = 1, 2, 3, . . . ) belong to an open convex set K of Rm and
that f (x) is a nonnegative convex function from K to R. Then the inequality
(2.1) holds for any measurable set Ω ⊆ Rn .
Proof. Put ΩR = Ω ∩ {w : |w| < R, w ∈ Rn }. Then ΩR is a bounded set in
n m
Rn for all fixed positive real numbers R. Since ui → u weakly* in (L∞
µ (R )) ,
∞
m
∞
1
ui → u weakly* in (Lµ (ΩR )) . Hence, by L (ΩR ) ⊂ L (ΩR ), it can be easily
shown that ui → u weakly in (L1µ (ΩR ))m . Then, using the process of the proof
of Theorem 2.1, we obtain
Z
Z
(3.1)
lim
f (ui )dµ ≥
f (u)dµ.
i→+∞
ΩR
ΩR
It follows from the nonnegativity of the convex function f that
Z
Z
(3.2)
lim
f (ui )dµ ≥
f (u)dµ.
i→+∞
Ω
ΩR
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Finally, by the Lebesgue monotonous convergence theorem, as R → +∞, (3.2)
implies (2.1). Our proof is completed.
Furthermore, by removing the nonnegativity of f (x) and assuming that the
convex set K is closed, we can deduce the following result.
∞
n m
Theorem 3.2. Assume that a sequence {ui }+∞
i=1 weakly* converges in (Lµ (R ))
to u as i → +∞, where m and n are two positive integers. Assume that all the
values of u and ui (i = 1, 2, 3, . . . ) belong to a closed convex set K in Rm and
that f (x) is a continuous convex function from K to R. Then the inequality (2.1)
holds for any bounded measurable set Ω ⊂ Rn .
We can also obtain Theorem 3.1 from Theorem 3.2. Theorem 3.2 can be
easily proved using Lemma 2.2. In fact, Theorem 3.2 is a part of the results
given by Ying [6]. However, we still give its proof below.
R
Proof Rof Theorem 3.2. Put αi = Ω f (ui )dµ (i = 1, 2, . . . ) and α =
lim Ω f (ui )dµ for any bounded set Ω. Then there exists a subsequence of
i→+∞
{αi }+∞
i=1
such that this subsequence, denoted without loss of generality by {αi }+∞
i=1 ,
converges to α as i → +∞.
e = K ∩ {x : x ∈ Rm , |x| ≤ kuk∞ + 1}. Then, since K is closed
Take K
e is a bounded closed set and f (x) is uniformly
and f (x) is continuous in K, K
e that is, for any given positive number ε, there exists a positive
continuous in K,
number δ < 1 such that
(3.3)
|f (x) − f (y)| < ε/µ(Ω)
e
as |x − y| < δ for any x and y in K.
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n m
∞
m
Since ui → u weakly* in (L∞
µ (R )) , ui → u weakly* in (Lµ (Ω)) .
Hence, by L∞ (Ω) ⊂ L1 (Ω), it can be easily shown that ui → u weakly in
(L1µ (Ω))m . It follows from Lemma 2.2 that, for any natural number j, there
PN (j) (j)
exists
a
convex
combination
k=j λk uk of {uk : k = j, j + 1, . . . } such that
PN (j) (j) 1
u − k=j λk uk ∞ ≤ j where N (j) is a natural number which depends on
m
j and {uk : k = j, j + 1, . . . }, kvk∞ represents a norm of v in (L∞
µ (Ω)) ,
P
P
(j)
N (j) (j)
N (j) (j)
λk ≥ 0 and k=j λk = 1. Put vj =
k=j λk uk . Then vj converges in
m
∞
(Lµ (Ω)) to u as j tends to ∞. In particular, for the above number δ, there
exists a natural number N such that
ess sup |u(ω) − vj (ω)| < δ
(3.4)
ω∈Ω
for all j > N. Since all the values of u and ui are in K and K is convex, all
e for all j > N. Combining (3.3) and (3.4),
the values of u and vj in Ω are in K
we know that for any given positive number ε, there exists a natural number N
such that
f (u) < f (vj ) + ε/µ(Ω)
(3.5)
almost everywhere in Ω for all j > N. Thus, integrating (3.5) gives
Z
Z
(3.6)
f (u)dµ ≤
f (vj )dµ + ε
Ω
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Ω
for all j > N. By the convexity of the function f (x), integrating (3.6) gives
Z
N (j)
X (j) Z
(3.7)
f (u)dµ ≤
λk
f (uk )dµ + ε.
Ω
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(3.7) can be equivalently written as
N (j)
Z
f (u)dµ ≤
(3.8)
Ω
X
(j)
λk αk + ε.
k=j
By letting j → +∞ and using the convergence of αk to α, (3.8) gives
Z
(3.9)
f (u)dµ ≤ α + ε.
Ω
By letting ε → 0, (3.9) leads to (2.1). This completes the proof.
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References
[1] R.J. DIPERNA AND P.L. LIONS, Global solutions of Boltzmann’s equation
and the entropy inequality, Arch. Rational Mech. Anal., 114 (1991), 47–55.
[2] J.J. BENEDETTO, Real Variable and Integration, B.G. Teubner, Stuttgart,
1976, p. 228–229.
[3] S.R. LAY, Convex Sets and Their Applications, John Wiley & Sons, Inc.,
New York, 1982, p. 214–215.
[4] R.T. ROCKAFELLAR, Convex Analysis, Princeton University Press,
1970.
[5] P. WOJTASZCZYK, Banach Spaces for Analysts, Cambridge University
Press, Cambridge, 1991, p. 28.
[6] YING LONGAN, Compensated compactness method and its application
to quasilinear hyperbolic equations, Advances In Mathematics (Chinese),
17(1) (1988).
[7] K. YOSIDA, Functional Analysis, Springer-Verlag, 1965.
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