Journal of Inequalities in Pure and
Applied Mathematics
SOME ASPECTS OF CONVEX FUNCTIONS AND THEIR APPLICATIONS
volume 2, issue 1, article 4,
2001.
JAMAL ROOIN
Institute for Advanced Studies in Basic Sciences,
P.O. Box 45195-159, Gava Zang,
Zanjan 45195, IRAN.
Received 13 April, 2000;
accepted 04 October 2000.
Communicated by: N.S. Barnett
and
Faculty of Mathematical Sciences and Computer Engineering,
University for Teacher Education,
599 Taleghani Avenue, Tehran 15614,
IRAN.
EMail: Rooin@iasbs.ic.ir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
008-00
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Abstract
In this paper we will study some aspects of convex functions and as applications
prove some interesting inequalities.
2000 Mathematics Subject Classification: 26D15, 39B62, 43A15
Key words: Convex Functions, Means, Lp-Spaces, Fubini’s Theorem
The author is supported in part by the Institute for Advanced Studies in Basic Sciences.
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
In [2] Sever S. Dragomir and Nicoleta M. Ionescu have studied some aspects
of convex functions and obtained some interesting inequalities. In this paper
we generalize the above paper to a very general case by introducing a suitable
convex function of a real variable from a given convex function. Studying its
properties leads to some remarkable inequalities in different abstract spaces.
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
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2.
The Main Results
The aim of this section is to study the properties of the function F defined below
as Theorems 2.2 and 2.5.
First we mention the following simple lemma, which describes the behavior
of a convex function defined on a closed interval of the real line.
Lemma 2.1. Let F be a convex function on the closed interval [a, b]. Then, we
have
(i) F takes its maximum at a or b.
(ii) F is bounded from below.
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
(iii) F (a+) and F (b−) exist (and are finite).
(iv) If the infimum of F over [a, b] is less than F (a+) and F (b−), then F takes
its minimum at a point x0 in (a, b).
(v) If a ≤ x0 < b (or a < x0 ≤ b), and F (x0 +) (or (F (x0 −)) is the infimum
of F over [a, b], then F is monotone decreasing on [a, x0 ] (or [a, x0 )) and
monotone increasing on (x0 , b] (or [x0 , b]).
Proof. See [3, 4].
Definition 2.1. Let X be a linear space, and f : C ⊆ X → R be a convex
mapping on a convex subset C of X. For n given elements x1 , x2 , · · · , xn of C,
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we define the following mapping of real variable F : [0, 1] → R by
!
n
n
X
P
f
aij (t)xj
i=1
F (t) =
j=1
n
,
where aij : [0, 1] → R+ (i, j = 1, · · · , n) are affine mappings, i.e., aij (αt1 +
βt2 ) = αaij (t1 ) + βaij (t2 ) for all α, β ≥ 0 with α + β = 1 and t1 , t2 in [0, 1],
and for each i and j
n
X
i=1
aij (t) = 1,
n
X
aij (t) = 1 (0 ≤ t ≤ 1).
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
j=1
The next theorem contains some remarkable properties of this mapping.
Theorem 2.2. With the above assumptions, we have:
f (x1 ) + · · · + f (xn )
x1 + · · · + xn
≤ F (t) ≤
(i) f
n
n
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(0 ≤ t ≤ 1).
(ii) F is convex on [0, 1].
R1
x1 + · · · + xn
f (x1 ) + · · · + f (xn )
(iii) f
≤ 0 F (t)dt ≤
.
n
n
Pn
(iv) Let pi ≥ 0 with Pn =
i=1 pi > 0, and ti are in [0, 1] for all i =
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1, 2, · · · , n. Then, we have the inequality:
!
n
x1 + · · · + xn
1 X
(2.1) f
≤F
pi ti
n
Pn i=1
n
1 X
f (x1 ) + · · · + f (xn )
≤
pi F (ti ) ≤
,
Pn i=1
n
which is a discrete version of Hadamard’s result.
Proof. (i) By the convexity of f , for all 0 ≤ t ≤ 1, we have
!
Pn Pn
i=1
j=1 aij (t)xj
F (t) ≥ f
n
!
Pn Pn
j=1
i=1 aij (t)xj
= f
n
!
Pn
j=1 xj
= f
,
n
and
Pn Pn
i=1
j=1 aij (t)f (xj )
F (t) ≤
Pn Pn n
j=1
i=1 aij (t)f (xj )
=
n
Pn
j=1 f (xj )
=
.
n
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
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(ii) Let α, β ≥ 0 with α + β = 1 and t1 , t2 be in [0, 1]. Then,
(2.2) F (αt1 + βt2 )
P
Pn
n
i=1 f
j=1 aij (αt1 + βt2 )xj
=
P n
Pn
Pn
n
f
α
a
(t
)x
+
β
a
(t
)x
i=1
j=1 ij 1 j
j=1 ij 2 j
=
n P
P
Pn
Pn
n
n
a
(t
)x
a
(t
)x
f
f
ij
1
j
ij
2
j
i=1
j=1
i=1
j=1
≤α
+β
n
n
= αF (t1 ) + βF (t2 ).
Thus F is convex.
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
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(iii) F being convex on [0, 1], is integrable on [0, 1], and by (i), we get (iii).
(iv) The first and last inequalities in (2.1) are obvious from (i), and the second
inequality follows from Jensen’s inequality applied for the convex function
F.
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Lemma 2.3. The general form of an affine mapping g : [0, 1] → R is
g(t) = (1 − t)k0 + tk1 ,
where k0 and k1 are two arbitrary real numbers.
The proof follows by considering t = (1 − t) · 0 + t · 1.
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Lemma 2.4. If aij : [0, 1]
R+ (i, j = 1, 2, P
· · · , n) are affine mappings such
P→
n
that for each t, i and j, i=1 aij (t) = 1 and nj=1 aij (t) = 1, then there exist
nonnegative numbers bij and cij , such that
(2.3)
aij (t) = (1 − t)bij + tcij (0 ≤ t ≤ 1; i, j = 1, · · · , n),
and for any i and j
n
X
bij =
i=1
n
X
cij = 1, and
i=1
n
X
bij =
j=1
n
X
cij = 1.
j=1
Proof. The decomposition of (2.3) is immediate from Lemma 2.3, and the rest
of the proof comes from below:
n
X
0 ≤ aij (0) = bij , 0 ≤ aij (1) = cij ,
n
n
n
X
X
X
bij =
aij (0) = 1,
cij =
aij (1) = 1,
i=1
i=1
i=1
i=1
n
X
n
X
n
X
n
X
j=1
bij =
j=1
aij (0) = 1,
cij =
j=1
aij (1) = 1.
j=1
Some Aspects of Convex
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Jamal Rooin
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Remark 2.1. A lot of simplifications occur if we take
(2.4)
bij = δij and cij = δi,n+1−j (i, j = 1, · · · , n),
in Lemma 2.4, where δij is the Kronecker delta.
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Theorem 2.5. With the above assumptions, if bij and cij are in the form (2.4),
then we have:
(i) For each t in 0, 12 , F 12 + t = F 12 − t .
(ii) max{F (t) : 0 ≤ t ≤ 1} = F (0) = F (1) = n1 (f (x1 ) + · · · + f (xn )).
P
(iii) min{F (t) : 0 ≤ t ≤ 1} = F 12 = ni=1 f xi +x2n+1−i n.
(iv) F is monotone decreasing on 0, 21 and monotone increasing on 12 , 1 .
Proof. (i) Since bij = δij and cij = δi,n+1−j , we have
n
P
F (t) =
(2.5)
Jamal Rooin
f [(1 − t)xi + txn+1−i ]
i=1
,
n
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and therefore, for each t in 0, 21 ,
F
1
−t
=
2
n
P
f
i=1
=
Contents
1
2
+ t xi +
1
2
− t xn+1−i
n
n
P
f
i=1
1
2
+ t xn+1−i +
n
= F
Some Aspects of Convex
Functions and Their
Applications
1
+t .
2
(ii) It is obvious from (2.5), and (i) of Lemma 2.1.
1
2
− t xi
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(iii) If F ( 12 ) is not the minimum of F over [0, 1], then by (i), there is a 0 < t ≤
1
, such that
2
1
1
1
F
−t =F
+t <F
.
2
2
2
But, using the convexity of F over [0, 1], we have
1
1
1
1
1
1
F
≤ F
−t + F
+t <F
,
2
2
2
2
2
2
a contradiction.
(iv) It is obvious from (iii) of Theorem 2.5, and (v) of Lemma 2.1.
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Jamal Rooin
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3.
Applications
Application 1. Let x1 , x2 , · · · , xn be n nonnegative numbers. Then, with the
above notations, we have
v
uY
n
u n X
√
x1 + x2 + · · · + xn
n
n
t
(3.1)
x1 x2 · · · xn ≤
[(1 − t)bij + tcij ]xj ≤
,
n
i=1 j=1
v
u n
uY
√
x1 + x2 + · · · + xn
n
n
x1 x2 · · · xn ≤ t
,
[(1 − t)xi + txn+1−i ] ≤
n
i=1
(3.2)
Some Aspects of Convex
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for all t in [0, 1], and
(3.3)
√
n
x1 x2 · · · xn
v
u 
Pj cij xj  P c x −1 P b x
u
P
( j ij j j ij j )
uY
n
u
j cij xj
n


−1 u
≤ e t 
Pj bij xj 
P
i=1
j bij xj
≤
x1 + x2 + · · · + xn
.
n
In particular
(3.4)
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√
n
x1 x2 · · · xn
v
u n xn+1−i 1
u
Y xn+1−i (xn+1−i −xi )
n
−1 t
≤ e
xxi i
i=1
≤
x1 + x2 + · · · + xn
,
n
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and
(3.5)
(3.6)
√
−1
x1 x2 ≤ e
2n + 2
2n + 1
1
1+
n
xx1 1
xx2 2
1
1 −x2
x
n
r
≤e≤
≤
x1 + x2
,
2
n+1
n
1
1+
n
n
.
Some Aspects of Convex
Functions and Their
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Proof. If we take f : (0, ∞) → R, f (x) = − ln x, then we have
!
n
n
X
1X
F (t) = −
ln
[(1 − t)bij + tcij ]xj ,
n i=1
j=1
and
Z 1
0
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n
1X
F (t)dt = −
n i=1
1
Z
ln
0

n
X
Contents
!
[(1 − t)bij + tcij ]xj
dt
JJ
J
j=1
Pn
1P
c x − n
b x
j=1 ij j
j=1 ij j
Pnj=1 cij xj  Pn
n
j=1 cij xj
1 Y

= − ln
 P
Pnj=1 bij xj 
n i=1
n
j=1 bij xj
+ 1,
which proves (3.1) and (3.3). In particular, if we take bij = δij and cij =
δi,n+1−j (i, j = 1, · · · , n), we obtain (3.2) and (3.4) from (3.1) and (3.3) respectively. The result (3.5) is immediate from (3.4). If we take x1 = n, x2 = n + 1
in (3.5), we get (3.6).
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Application 2. If X is a Lebesgue measurable subset of Rk , p ≥ 1, and
f1 , f2 , · · · , fn belong to Lp = Lp (X), then we have
i
Pn hPn
Pn
p
c
|f
|;
b
|f
|
f1 + · · · + fn i=1
j=1 ij j
j=1 ij j
≤
(3.7)
n
n(p + 1)
p
≤
kf1 kpp + · · · + kfn kpp
,
n
and
(3.8)
Pn
f1 + · · · + fn p
i=1 [|fi | ; |fn+1−i |]
≤
n
n(p + 1)
p
≤
kf1 kpp + · · · + kfn kpp
,
n
where for each Lebesgue measurable function g ≥ 0 and h ≥ 0 on X,
p+1
Z p+1
p+1 g
−
h
g
− hp+1
[g; h] = =
dx,
g − h 1
g−h
X
when g(x) = h(x), the integrand is understood to be (p + 1)g p (x).
In particular, if p is an integer then,
(3.9)
f1 + · · · + fn p
≤
n
p
p n P
P
k p−k fi .fn+1−i i=1 k=0
n(p + 1)
1
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kf1 kpp + · · · + kfn kpp
≤
,
n
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and
(3.10)
f1 + f2 p
2 ≤
p
p P
k p−k f
.f
1 2 1
k=0
p+1
kf1 kpp + kf2 kpp
≤
,
2
Proof. Since
(f1 + · · · + fn ) (|f1 | + · · · + |fn |) ≤
n
n
p
p
and the Lp −norms of fi and |fi | are equal (i = 1, · · · , n), it is sufficient to
assume fi ≥ 0 (i = 1, · · · , n). If we take ϕ → kϕkp for the convex function
Lp → R, then using Fubini’s theorem we get
p
Z 1
n Z 1 X
n
X
1
F (t)dt =
[(1 − t)bij + tcij ]fj dt
n i=1 0 j=1
0
p
!p
Z
Z
n
n
1
X
X
1
=
[(1 − t)bij + tcij ]fj (x) dxdt
n i=1 0 X j=1
!p
n Z Z 1
n
X
1X
=
[(1 − t)bij + tcij ]fj (x) dtdx
n i=1 X 0
j=1
P
p+1 P
p+1
n
n
n Z
c
f
−
b
f
X
ij
j
ij
j
j=1
j=1
1
Pn
Pn
=
dx
n(p + 1) i=1 X
j=1 cij fj −
j=1 bij fj
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Pn hPn
=
i=1
j=1 cij fj ;
Pn
n(p + 1)
j=1 bij fj
i
,
which yields (3.7). In particular, if we set bij = δij and cij = δi,n+1−j (i, j =
1, · · · , n), (3.8) follows from (3.7). Finally, (3.9) and (3.10) are immediate from
(3.8).
Remark 3.1. Let X be a Lebesgue measurable subset of Rk with finite measure,
and M be the vector space of all Lebesgue measurable functions on X with
pointwise operations [1]. The set C, consisting of all nonnegative measurable
t
functions on X, is a convex subset of M. Since the function t → 1+t
(t ≥ 0) is
concave, the mapping ϕ : C → R with
Z
f
ϕ(f ) =
dx
(f ∈ C)
X 1+f
Some Aspects of Convex
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is concave.
Application 3. With the above notations, if f1 , · · · , fn belong to M, then
n
1X
(3.11)
n i=1
Z
X
|fi |
dx
1 + |fi |
P
n Z
1 + nj=1 cij |fj |
1X
1
P
P
≤ m(X) −
ln
dx
n i=1 X nj=1 (cij − bij )|fj | 1 + nj=1 bij |fj |
Pn
Z
1
i=1 |fi |
n
P
≤
dx,
n
1
X 1+ n
i=1 |fi |
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n
1X
n i=1
(3.12)
Z
X
|fi |
dx
1 + |fi |
n Z
1X
1
1 + |fn+1−i |
≤ m(X) −
ln
dx
n i=1 X |fn+1−i | − |fi |
1 + |fi |
Pn
Z
1
i=1 |fi |
n
P
≤
dx,
n
1
X 1+ n
i=1 |fi |
(3.13)
1
2
2 Z
X
i=1
X
|fi |
1
1 + |f2 |
dx ≤ m(X) −
ln
dx
1 + |fi |
1 + |f1 |
X |f2 | − |f1 |
P2
Z
1
i=1 |fi |
2
≤
dx,
P
2
1
X 1+ 2
i=1 |fi |
Z
Some Aspects of Convex
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in which, generally, when a = b > 0, the ratio (ln b−ln a)(b−a) is understood
as 1a.
Proof. We can suppose that fi ≥ 0 (1 ≤ i ≤ n). Since ϕ is concave, taking ϕ
and φ instead of f and F in Theorem 2.2 respectively, we get
Z 1
ϕ(f1 ) + · · · + ϕ(fn )
(3.14)
≤
φ(t)dt
n
0
f1 + · · · + fn
.
≤ ϕ
n
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However, by Fubini’s theorem and applying the change of variables
u=
n
X
[(1 − t)bij + tcij ]fj (x),
j=1
in the following integrals, we have,
Pn
Z
Z 1
n Z
1X 1
j=1 [(1 − t)bij + tcij ]fj (x)
P
φ(t)dt =
dxdt
n i=1 0 X 1 + nj=1 [(1 − t)bij + tcij ]fj (x)
0
Pn
n Z Z 1
1X
j=1 [(1 − t)bij + tcij ]fj (x)
P
dtdx
=
n i=1 X 0 1 + nj=1 [(1 − t)bij + tcij ]fj (x)
Z Pnj=1 cij fj (x) n Z
1X
1
1
P
=
1−
dudx
n i=1 X nj=1 (cij − bij )fj (x) Pnj=1 bij fj (x)
1+u
P
n Z
1 + nj=1 cij fj
1X
1
P
P
= m(X) −
dx,
ln
n i=1 X nj=1 (cij − bij )fj 1 + nj=1 bij fj
and after substituting this in (3.14), we obtain (3.11). The inequalities (3.12)
and (3.13) are special cases of (3.11), taking bij = δij and cij = δi,n+1−j .
Acknowledgement 1. I would like to express my gratitude to Professors A. R.
Medghalchi and B. Mehri for their valuable comments and suggestions.
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References
[1] S. BERBERIAN, Lectures in functional analysis and operator theory,
Springer, New York-Heidelberg-Berlin, 1974.
[2] S. S. DRAGOMIR AND N. M. IONESCU, Some remarks on convex functions, Revue d’analyse numérique et de théorie de l’approximation, 21
(1992), 31–36.
[3] A.W. ROBERTS AND D.E. VARBERG, Convex functions, Academic Press,
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[4] R. WEBSTER, Convexity, Oxford University Press, Oxford, New York,
Tokyo, 1994.
Some Aspects of Convex
Functions and Their
Applications
Jamal Rooin
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J. Ineq. Pure and Appl. Math. 2(1) Art. 4, 2001
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