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volume 2, issue 1, article 4, 2001.

Received 13 April, 2000;

accepted 04 October 2000.

Communicated by:N.S. Barnett

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Journal of Inequalities in Pure and Applied Mathematics

SOME ASPECTS OF CONVEX FUNCTIONS AND THEIR APPLICA- TIONS

JAMAL ROOIN

Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-159, Gava Zang,

Zanjan 45195, IRAN.

and

Faculty of Mathematical Sciences and Computer Engineering, University for Teacher Education,

599 Taleghani Avenue, Tehran 15614, IRAN.

EMail:Rooin@iasbs.ic.ir

c

2000Victoria University ISSN (electronic): 1443-5756 008-00

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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

http://jipam.vu.edu.au

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 Sci- ences.

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.

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2. The Main Results

The aim of this section is to study the properties of the functionF defined below as Theorems2.2and2.5.

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. LetF be a convex function on the closed interval[a, b]. Then, we have

(i) F takes its maximum ataorb.

(ii) F is bounded from below.

(iii) F(a+)andF(b−)exist (and are finite).

(iv) If the infimum ofF over[a, b]is less thanF(a+)andF(b−), thenF takes its minimum at a pointx₀in(a, b).

(v) Ifa≤ x₀ < b(ora < x₀ ≤ b), andF(x₀+)(or(F(x₀−)) is the infimum ofF over[a, b], thenF 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 subsetCofX. Forngiven elementsx₁, x₂,· · · , x_nofC,

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we define the following mapping of real variableF : [0,1]→Rby

F(t) =

n

P

i=1

f

n

X

j=1

a_ij(t)x_j

!

n ,

where aij : [0,1] → R⁺ (i, j = 1,· · · , n)are affine mappings, i.e., aij(αt1 + βt₂) = αa_ij(t₁) +βa_ij(t₂)for allα, β ≥0withα+β = 1andt₁, t₂ in[0,1], and for eachiandj

n

X

i=1

a_ij(t) = 1,

n

X

j=1

a_ij(t) = 1 (0 ≤t≤1).

The next theorem contains some remarkable properties of this mapping.

Theorem 2.2. With the above assumptions, we have:

(i) f

x₁+· · ·+x_n n

≤F(t)≤ f(x₁) +· · ·+f(x_n)

n (0≤t≤1).

(ii) F is convex on[0,1].

(iii) f

x₁+· · ·+x_n n

≤R1

0 F(t)dt≤ f(x₁) +· · ·+f(x_n)

n .

(iv) Let p_i ≥ 0 with P_n = Pn

i=1p_i > 0, and t_i are in [0,1] for all i =

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1,2,· · · , n. Then, we have the inequality:

f

x₁ +· · ·+x_n n

≤F 1

P_n

n

X

i=1

piti

! (2.1)

≤ 1 P_n

n

X

i=1

p_iF(t_i)≤ f(x₁) +· · ·+f(x_n)

n ,

which is a discrete version of Hadamard’s result.

Proof. (i) By the convexity off, for all0≤t≤1, we have

F(t) ≥ f Pn

i=1

Pn

j=1a_ij(t)x_j n

!

= f Pn

j=1

Pn

i=1a_ij(t)x_j n

!

= f Pn

j=1x_j n

! ,

and

F(t) ≤ Pn

i=1

Pn

j=1a_ij(t)f(x_j) n

= Pn

j=1

Pn

i=1a_ij(t)f(x_j) n

= Pn

j=1f(x_j)

n .

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(ii) Letα, β ≥0withα+β = 1andt₁,t₂be in[0,1]. Then, F(αt₁+βt₂)

(2.2)

= Pn

i=1f Pn

j=1a_ij(αt₁+βt₂)x_j n

= Pn

i=1f

αPn

j=1aij(t1)xj +βPn

j=1aij(t2)xj

n

≤α Pn

i=1f Pn

j=1a_ij(t₁)x_j

n +β

Pn i=1f

Pn

j=1a_ij(t₂)x_j n

=αF(t₁) +βF(t₂).

ThusF is convex.

(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.

Lemma 2.3. The general form of an affine mappingg : [0,1]→Ris g(t) = (1−t)k₀+tk₁,

wherek₀ andk₁ are two arbitrary real numbers.

The proof follows by consideringt= (1−t)·0 +t·1.

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Lemma 2.4. Ifa_ij : [0,1] →R⁺ (i, j = 1,2,· · · , n)are affine mappings such that for eacht, iandj,Pn

i=1aij(t) = 1andPn

j=1aij(t) = 1, then there exist nonnegative numbersb_ij andc_ij, such that

(2.3) a_ij(t) = (1−t)b_ij +tc_ij (0≤t ≤1; i, j = 1,· · · , n), and for anyiandj

n

X

i=1

b_ij =

n

X

i=1

c_ij = 1, and

n

X

j=1

b_ij =

n

X

j=1

c_ij = 1.

Proof. The decomposition of (2.3) is immediate from Lemma2.3, and the rest of the proof comes from below:

0 ≤ a_ij(0) =b_ij, 0≤a_ij(1) =c_ij,

n

X

i=1

bij =

n

X

i=1

aij(0) = 1,

n

X

i=1

cij =

n

X

i=1

aij(1) = 1,

n

X

j=1

bij =

n

X

j=1

aij(0) = 1,

n

X

j=1

cij =

n

X

j=1

aij(1) = 1.

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 Lemma2.4, whereδij is the Kronecker delta.

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Theorem 2.5. With the above assumptions, if b_ij and c_ij are in the form (2.4), then we have:

(i) For eachtin 0,¹₂

,F ¹₂ +t

=F ¹₂ −t .

(ii) max{F(t) : 0 ≤t≤1}=F(0) =F(1) = _n¹(f(x₁) +· · ·+f(x_n)).

(iii) min{F(t) : 0≤t ≤1}=F ¹₂

=Pn

i=1f ^xⁱ^+xⁿ⁺¹⁻ⁱ₂ n.

(iv) F is monotone decreasing on 0,¹₂

and monotone increasing on₁

2,1 . Proof. (i) Sincebij =δij andcij =δi,n+1−j, we have

(2.5) F(t) =

n

P

i=1

f[(1−t)x_i+tx_n+1−i]

n ,

and therefore, for eachtin 0,¹₂

,

F 1

2−t

=

n

P

i=1

f ₁

2 +t

x_i+ ¹₂ −t

xn+1−i

n

=

n

P

i=1

f ₁

2 +t

xn+1−i+ ¹₂ −t x_i n

= F 1

2 +t

.

(ii) It is obvious from (2.5), and (i) of Lemma2.1.

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(iii) IfF(¹₂)is not the minimum ofF over[0,1], then by (i), there is a0< t≤

1

2, such that

F 1

2 −t

=F 1

2+t

< F 1

2

. But, using the convexity ofF over[0,1], we have

F 1

2

≤ 1 2F

1 2 −t

+ 1

2F 1

2+t

< F 1

2

, a contradiction.

(iv) It is obvious from (iii) of Theorem2.5, and (v) of Lemma2.1.

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3. Applications

Application 1. Let x₁, x₂,· · · , x_n be n nonnegative numbers. Then, with the above notations, we have

(3.1) √ⁿ

x₁x₂· · ·x_n ≤ ⁿ v u u t

n

Y

i=1 n

X

j=1

[(1−t)b_ij +tc_ij]x_j ≤ x₁+x₂ +· · ·+x_n

n ,

(3.2) √ⁿ

x₁x₂· · ·x_n ≤ ⁿ v u u t

n

Y

i=1

[(1−t)x_i+txn+1−i]≤ x1+x2+· · ·+xn

n ,

for alltin[0,1], and

√n

x₁x₂· · ·x_n ≤ e⁻¹ ⁿ v u u u u u t

n

Y

i=1





 P

jcijxj

^P_jcijxj

P

jb_ijx_j^P_jbijxj







1

(^Pj cij xj−P j bij xj) (3.3)

≤ x₁+x₂+· · ·+x_n

n .

In particular

√n

x₁x₂· · ·x_n ≤ e⁻¹ ⁿ v u u t

n

Y

i=1

x^x_n+1−iⁿ⁺¹⁻ⁱ x^x_iⁱ

₍ ¹

xn+1−i−xi)

(3.4)

≤ x1 +x2+· · ·+xn

n ,

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and

(3.5) √

x₁x₂ ≤e⁻¹ x^x₁¹

x^x₂² _x ¹

1−x2

≤ x₁+x₂ 2 ,

(3.6) 2n+ 2

2n+ 1

1 + 1 n

n

≤e≤

rn+ 1 n

1 + 1

n n

.

Proof. If we takef : (0,∞)→R, f(x) =−lnx,then we have

F(t) = −1 n

n

X

i=1

ln

n

X

j=1

[(1−t)b_ij+tc_ij]x_j

! ,

and Z 1

0

F(t)dt = −1 n

n

X

i=1

Z 1 0

ln

n

X

j=1

[(1−t)b_ij +tc_ij]x_j

! dt

= −1 n ln

n

Y

i=1





 Pn

j=1cijxj

^Pⁿ_j=1cijxj

Pn

j=1b_ijx_j^Pⁿ_j=1bijxj







1 Pn

j=1cij xj−Pn j=1bij 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 takex₁ =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 R^k, p ≥ 1, and f1, f2,· · · , fnbelong toL^p =L^p(X), then we have

f1+· · ·+fn

n

p

≤ Pn

i=1

hPn

j=1c_ij|f_j|;Pn

j=1b_ij|f_j|i n(p+ 1)

(3.7)

≤ kf₁k^p_p+· · ·+kf_nk^p_p

n ,

and

f₁+· · ·+f_n n

p

≤ Pn

i=1[|f_i|;|fn+1−i|]

n(p+ 1) (3.8)

≤ kf1k^p_p+· · ·+kfnk^p_p

n ,

where for each Lebesgue measurable functiong ≥0andh≥0onX,

[g;h] =

g^p+1−h^p+1 g−h

1

= Z

X

g^p+1−h^p+1 g−h dx, wheng(x) =h(x), the integrand is understood to be(p+ 1)g^p(x).

In particular, ifpis an integer then,

(3.9)

f₁ +· · ·+f_n n

p

≤

n

P

i=1 p

P

k=0

f_i^k.f_n+1−i^p−k 1

n(p+ 1) ≤ kf₁k^p_p+· · ·+kf_nk^p_p

n ,

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and

(3.10)

f₁ +f₂ 2

p

≤

p

P

k=0

f₁^k.f₂^p−k 1

p+ 1 ≤ kf₁k^p_p+kf₂k^p_p

2 ,

Proof. Since

(f₁+· · ·+f_n) n

p

≤

(|f₁|+· · ·+|f_n|) n

p

and the L^p−norms of fi and |fi| are equal (i = 1,· · · , n), it is sufficient to assume f_i ≥ 0 (i = 1,· · · , n). If we take ϕ → kϕk^p for the convex function L^p →R, then using Fubini’s theorem we get

Z 1 0

F(t)dt = 1 n

n

X

i=1

Z 1 0

n

X

j=1

[(1−t)b_ij +tc_ij]f_j

p

dt

= 1 n

n

X

i=1

Z 1 0

Z

X n

X

j=1

[(1−t)b_ij+tc_ij]f_j(x)

!p

dxdt

= 1 n

n

X

i=1

Z

X

Z 1 0

n

X

j=1

[(1−t)b_ij+tc_ij]f_j(x)

!p

dtdx

= 1

n(p+ 1)

n

X

i=1

Z

X

Pn

j=1c_ijf_jp+1

− Pn

j=1b_ijf_jp+1

Pn

j=1c_ijf_j−Pn

j=1b_ijf_j dx

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= Pn

i=1

hPn

j=1cijfj;Pn j=1bijfj

i

n(p+ 1) ,

which yields (3.7). In particular, if we set bij = δij andcij = 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. LetXbe a Lebesgue measurable subset ofR^kwith finite measure, and M be the vector space of all Lebesgue measurable functions on X with pointwise operations [1]. The setC, consisting of all nonnegative measurable functions onX, is a convex subset ofM. Since the functiont → _1+t^t (t ≥0)is concave, the mappingϕ:C →Rwith

ϕ(f) = Z

X

f

1 +fdx (f ∈C)

is concave.

Application 3. With the above notations, iff₁,· · · , f_nbelong toM, then 1

n

X

i=1

Z

X

|f_i| 1 +|fi|dx (3.11)

≤m(X)− 1 n

n

X

i=1

Z

X

1 Pn

j=1(c_ij −b_ij)|f_j|ln1 +Pn

j=1cij|fj| 1 +Pn

j=1b_ij|f_j|dx

≤ Z

X 1 n

Pn i=1|f_i| 1 + _n¹ Pn

i=1|f_i|dx,

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1 n

n

X

i=1

Z

X

|f_i| 1 +|fi|dx (3.12)

≤m(X)− 1 n

n

X

i=1

Z

X

1

|f_n+1−i| − |f_i|ln1 +|fn+1−i| 1 +|f_i| dx

≤ Z

X 1 n

Pn i=1|f_i| 1 + ¹_nPn

i=1|f_i|dx,

1 2

2

X

i=1

Z

X

|f_i|

1 +|f_i|dx ≤ m(X)− Z

X

1

|f₂| − |f₁|ln1 +|f₂| 1 +|f₁|dx (3.13)

≤ Z

X 1 2

P2 i=1|f_i| 1 + ¹₂P2

i=1|f_i|dx,

in which, generally, whena=b >0, the ratio(lnb−lna)(b−a)is understood as1a.

Proof. We can suppose thatf_i ≥0 (1≤ i≤ n). Sinceϕis concave, takingϕ andφinstead off andF in Theorem2.2respectively, we get

ϕ(f1) +· · ·+ϕ(fn)

n ≤

Z 1 0

φ(t)dt (3.14)

≤ ϕ

f₁+· · ·+f_n n

.

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However, by Fubini’s theorem and applying the change of variables

u=

n

X

j=1

[(1−t)bij +tcij]fj(x),

in the following integrals, we have,

Z 1 0

φ(t)dt = 1 n

n

X

i=1

Z 1 0

Z

X

Pn

j=1[(1−t)b_ij +tc_ij]f_j(x) 1 +Pn

j=1[(1−t)b_ij +tc_ij]f_j(x)dxdt

= 1 n

n

X

i=1

Z

X

Z 1 0

Pn

j=1[(1−t)b_ij +tc_ij]f_j(x) 1 +Pn

j=1[(1−t)b_ij +tc_ij]f_j(x)dtdx

= 1 n

n

X

i=1

Z

X

1 Pn

j=1(c_ij −b_ij)f_j(x)

Z ^Pⁿ_j=1cijfj(x) Pn

j=1bijfj(x)

1− 1 1 +u

dudx

= m(X)− 1 n

n

X

i=1

Z

X

1 Pn

j=1(cij −bij)fj

ln1 +Pn j=1c_ijf_j 1 +Pn

j=1bijfj

dx,

and after substituting this in (3.14), we obtain (3.11). The inequalities (3.12) and (3.13) are special cases of (3.11), takingb_ij =δ_ij andc_ij =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 ANDN. M. IONESCU, Some remarks on convex func- tions, Revue d’analyse numérique et de théorie de l’approximation, 21 (1992), 31–36.

[3] A.W. ROBERTSANDD.E. VARBERG, Convex functions, Academic Press, New York and London, 1973.

[4] R. WEBSTER, Convexity, Oxford University Press, Oxford, New York, Tokyo, 1994.