Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 184, 2006
CONVEX FUNCTIONS AND INEQUALITIES FOR INTEGRALS
ZHENGLU JIANG, XIAOYONG FU, AND HONGJIONG TIAN
D EPARTMENT OF M ATHEMATICS
Z HONGSHAN U NIVERSITY
G UANGZHOU 510275, C HINA
mcsjzl@mail.sysu.edu.cn
D EPARTMENT OF M ATHEMATICS
Z HONGSHAN U NIVERSITY
G UANGZHOU 510275, C HINA
mcsfxy@mail.sysu.edu.cn
D EPARTMENT OF M ATHEMATICS
S HANGHAI T EACHERS ’ U NIVERSITY
S HANGHAI 200234, C HINA
hongjiongtian@263.net
Received 04 June, 2005; accepted 01 December, 2006
Communicated by S.S. Dragomir
A BSTRACT. 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 vectorvalued 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.
Key words and phrases: Convex functions; Measure spaces; Lebesgue integrals; Banach spaces.
2000 Mathematics Subject Classification. Primary 26D15.
1. I NTRODUCTION
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
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
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.
178-05
2
Z HENGLU J IANG , X IAOYONG F U ,
AND
H ONGJIONG T IAN
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
p1
p
(1.1)
kukp =
|u(x)| dµ
Rn
(Lpµ (Rn ))m
for any p ∈ [1, +∞),
denotes a Banach vector space where each measurable vectorn
valued function has m components in Lpµ (Rn ), L∞
µ (R ) represents a Banach space where each
measurable function u(x) has the following 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
n
space where each measurable vector-valued function has m components in L∞
µ (R ).
p
n
Below, we give the definition of weak convergence of a sequence in Lµ (R ), where p ∈
[1, +∞). Assume that q = p/(p − 1) as p ∈ (1, +∞) and that q = ∞ as p = 1. If u ∈ Lpµ (Rn )
p
n
and it is assumed that a sequence {ui }+∞
i=1 in Lµ (R ) satisfies
Z
Z
(1.3)
lim
ui vdµ =
uvdµ
i→+∞
Lqµ (Rn ),
Rn
Rn
{ui }+∞
i=1
for all v ∈
then the sequence
is said to be weakly convergent in Lpµ (Rn ) to
u as i → +∞. Similarly, we introduce the definition of weak* convergence of a sequence in
+∞
n
n
∞
n
∞
L∞
µ (R ). If u ∈ Lµ (R ) and it is assumed that a sequence {ui }i=1 in Lµ (R ) satisfies the
equality (1.3) for all v ∈ L1µ (Rn ), then the sequence {ui }+∞
i=1 is said to be weakly* convergent
∞
n
in Lµ (R ) to u as i → +∞. Then we define the weak (or weak*) convergence of a sequence
+∞
n m
p
n
in (Lpµ (Rn ))m (or (L∞
µ (R )) ). If 1 ≤ p < +∞ and {uji }i=1 is weakly convergent in Lµ (R )
+∞
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 space (Lpµ (Rn ))m to û = (û1 , û2 , . . . , ûm ) as
∞
n
i → +∞. Similarly, if {uji }+∞
i=1 is weakly* convergent in Lµ (R ) to ûj for all j = 1, 2, . . . , m
+∞
as i → +∞, then a sequence {ui = (u1i , u2i , . . . , umi )}i=1 is said to be weakly* convergent in
n m
a Banach vector space (L∞
µ (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
n m
on a vector space Rm and vector-valued functions in a weakly* 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
J. Inequal. Pure and Appl. Math., 7(5) Art. 184, 2006
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C ONVEX F UNCTIONS AND I NEQUALITIES FOR I NTEGRALS
3
defined by F (x, y) = (x − y) log( xy ) for (x, y) ∈ (0, +∞) × (0, ∞) and {(ai , bi )}+∞
i=1 is a
1
n 2
nonnegative sequence weakly 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 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.
2. I NEQUALITIES FOR W EAKLY C ONVERGENT S EQUENCES
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
f (ui )dµ ≥
f (u)dµ
(2.1)
lim
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 → P
u weakly in a normed
space. Then there exists, for any
Plinear
n
n
ε > 0, a convex
combination k=1 λk uk (λk ≥ 0, k=1 λk = 1) of {uk : k = 1, 2, . . . } such
P
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 Rbe a bounded set defined by ΩR = Ω ∩ {w
R : |w| < R, w ∈
n
R } for all R > 0. Put αi = ΩR f (ui )dµ (i = 1, 2, . . . ) and α = lim ΩR f (ui )dµ for all
i→+∞
{αi }+∞
i=1
the bounded sets ΩR . Then there exists a subsequence of
such that this subsequence,
denoted without loss of generality by {αi }+∞
,
converges
to
α
as
i
→
+∞.
i=1
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 and Kl ⊂ Kl+1 for any positive integer l. It can also be easily proven that
l→+∞
lim Dl (u) = Rn and that Dl (u) ⊆ Dl+1 (u) for any positive integer l. Since f (x) is a convex
l→+∞
function defined on an open convex set K, f (x) is continuous in K (see [3]). Thus f (x) is
J. Inequal. Pure and Appl. Math., 7(5) Art. 184, 2006
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4
Z HENGLU J IANG , X IAOYONG F U ,
AND
H ONGJIONG T IAN
uniformly continuous in Kl+1 , that is, for any given positive number ε, there exists a positive
number δ such that
|f (x) − f (y)| < ε/µ(ΩR )
(2.2)
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 that for any
PN (j)
natural number j, there exists a convex combination k=j λk uk of {uk : k = j, j + 1, . . . }
PN (j) (j)
such that u − k=j λk uk p ≤ 1j , where N (j) is a natural number which depends on j and
PN (j) (j)
(j)
{uk : k = j, j+1, . . . }, kvkp represents a norm of v in (Lpµ (Rn ))m , λk ≥ 0 and k=j λk = 1.
PN (j) (j)
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 generality
+∞
by {vj }j=1 , converges almost everywhere in Rn to u as j goes to 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
f (u) < f (vj ) + ε/µ(ΩR )
(2.4)
in (ΩR \ER ) ∩ Dl (u) for all j > N. Since all the values of {ui }+∞
i=1 and u are in K and f (x) is
convex and nonnegative, integrating (2.4) gives
Z
N (j)
X (j) Z
λk
f (uk )dµ + σ max f (x) + ε.
(2.5)
f (u)dµ ≤
ΩR ∩Dl (u)
x∈Kl
ΩR
k=j
(2.5) can be equivalently written as
Z
N (j)
X (j)
λk αk + σ max f (x) + ε.
(2.6)
f (u)dµ ≤
ΩR ∩Dl (u)
x∈Kl
k=j
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 ΩR . By first letting
l→+∞
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.
J. Inequal. Pure and Appl. Math., 7(5) Art. 184, 2006
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C ONVEX F UNCTIONS AND I NEQUALITIES FOR I NTEGRALS
5
3. I NEQUALITIES FOR W EAKLY * C ONVERGENT S EQUENCES
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
n m
similar result for weakly* convergent sequences in (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 }+∞
to u as
i=1 weakly* converges in (Lµ (R ))
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 Rn for all
n m
fixed positive real numbers R. Since ui → u weakly* in (L∞
µ (R )) , ui → u weakly* in
m
∞
1
(L∞
µ (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
f (ui )dµ ≥
f (u)dµ.
(3.1)
lim
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
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
R
Proof of Theorem 3.2. Put αi = Ω f (ui )dµ (i = 1, 2, . . . ) and α = lim Ω f (ui )dµ for any
i→+∞
bounded set Ω. 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 → +∞.
e = K ∩ {x : x ∈ Rm , |x| ≤ kuk∞ + 1}. Then, since K is closed and f (x) is
Take K
e is a bounded closed set and f (x) is uniformly continuous in K,
e that is, for
continuous in K, K
any given positive number ε, there exists a positive 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.
n m
∞
m
∞
Since ui → u weakly* in (L∞
µ (R )) , ui → u weakly* in (Lµ (Ω)) . Hence, by L (Ω) ⊂
1
1
m
L (Ω), it can be easily shown that ui → u weakly in (Lµ (Ω)) . It follows from Lemma 2.2
PN (j) (j)
that, for any natural number j, there exists a convex combination k=j λk uk of {uk : k =
PN (j) (j)
j, j + 1, . . . } such that u − k=j λk uk ∞ ≤ 1j where N (j) is a natural number which
J. Inequal. Pure and Appl. Math., 7(5) Art. 184, 2006
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6
Z HENGLU J IANG , X IAOYONG F U ,
AND
H ONGJIONG T IAN
(j)
m
depends on j and {uk : k = j, j + 1, . . . }, kvk∞ represents a norm of v in (L∞
µ (Ω)) , λk ≥ 0
PN (j) (j)
PN (j) (j)
m
and k=j λk = 1. Put vj = k=j λk uk . Then vj converges in (L∞
µ (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 the values of u and
e for all j > N. Combining (3.3) and (3.4), we know that for any given positive
vj in Ω are in K
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
Ω
for all j > N. By the convexity of the function f (x), integrating (3.6) gives
Z
N (j)
X (j) Z
λk
f (uk )dµ + ε.
(3.7)
f (u)dµ ≤
Ω
Ω
k=j
(3.7) can be equivalently written as
Z
N (j)
X (j)
f (u)dµ ≤
λk αk + ε.
(3.8)
Ω
k=j
By letting j → +∞ and using the convergence of αk to α, (3.8) gives
Z
f (u)dµ ≤ α + ε.
(3.9)
Ω
By letting ε → 0, (3.9) leads to (2.1). This completes the proof.
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J. Inequal. Pure and Appl. Math., 7(5) Art. 184, 2006
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