Volume 9, Number 4 (2015), 12451257 doi:10.7153jmi-09-95 [614014]

Journal of
Mathematical
Inequalities
Volume 9, Number 4 (2015), 1245–1257 doi:10.7153/jmi-09-95
CONVEX ORDERING PROPERTIES AND APPLICATIONS
AURELIA FLOREA , EUGEN P˘ALT˘ANEA AND DUMITRU B˘AL˘A
(Communicated by M. Mati´ c )
Abstract. A relevant application of the stochastic convex order is the well-known weighted
Hermite-Hadamard inequality, where the weight is provided by a given probability distribu-
tion. Our goal is to show that, starting from such a fixed weigh t, we can fill the whole space
between the Hermite-Hadamard bounds by highlighting some p arametric families of probabil-
ity distributions. Thus, we propose two alternative constr uctions based on the convex ordering
properties.
1. Introduction
The paper refers to the following result of Hermite (1883) an d Hadamard (1893).
THEOREM 1.If f:[a,b]→Ris a convex function, then
f/parenleftbigga+b
2/parenrightbigg
/lessorequalslant1
b−a/integraldisplayb
af(x)dx/lessorequalslantf(a)+f(b)
2. (1)
A rich literature has been stimulated by this result. We ment ion here a brief se-
lection of relevant extensions. Fej´ er [ 4] highlights in 1906 the first weighted version of
(1).
THEOREM 2.If f:[a,b]→Ris convex and g :[a,b]→[0,∞)is integrable and
symmetric about (a+b)/2, i.e. g (a+b−x) =g(x),∀x∈(a,b), with/integraltextb
ag(x)dx>0,
then
f/parenleftbigga+b
2/parenrightbigg
/lessorequalslant/integraltextb
af(x)g(x)dx
/integraltextb
ag(x)dx/lessorequalslantf(a)+f(b)
2. (2)
Remark that for g(x) =1,∀x∈[a,b], we obtain the Hermite-Hadamard inequal-
ity. Another interesting generalization is due to Brenner a nd Alzer [ 2].
THEOREM 3.If f:[a,b]→Ris convex and g :[a,b]→[0,∞)is integrable and
symmetric about c ∈(a,b), such that/integraltextc+t
c−tg(x)dx>0,∀0/lessorequalslantt/lessorequalslantmin{c−a,b−c},
then
f(c)/lessorequalslant/integraltextc+t
c−tf(x)g(x)dx
/integraltextc+t
c−tg(x)dx/lessorequalslantb−c
b−af(a)+c−a
b−af(b). (3)
Mathematics subject classification (2010): 26A51, 26B25, 26D10.
Keywords and phrases : Convex functions, Hermite-Hadamard inequality, convex o rder.
c/circlecopyrt /BW /D0 , Zagreb
Paper JMI-09-951245

1246 A. F LOREA , E. P ˘ALT˘ANEA AND D. B ˘AL˘A
Mention that Bakula, Peˇ cari´ c and Peri´ c have recently pro vided an extension of
Theorem 3for positive linear functionals (see [ 1]).
Fink proves in [ 5] a general weighted version of the Hermite-Hadamard inequa lity.
THEOREM 4.Letµbe a real Borel measure on [a,b], with µ([a,b])>0.
If f:[a,b]→Ris convex, then (under some restrictions of positivity on µ),
f/parenleftbig
xµ/parenrightbig
/lessorequalslant1
µ([a,b])/integraldisplayb
af(x)dµ(x)/lessorequalslantb−xµ
b−af(a)+xµ−a
b−af(b), (4)
where x µ=/integraltextb
axdµ(x)
µ([a,b])is the barycenter of µ.
Florea and Niculescu find in [ 6] a complete characterization of the measures µ
satisfying the right inequality of ( 4). A comprehensive treatment of convex functions
can be found, for example, in [ 8].
In this paper, we look at the weighted version ( 4) of the classical inequality of
Hermite and Hadamard from the perspective provided by the st ochastic convex order.
This approach is mainly due to Cal and C´ arcamo. Thus, in the p aper [ 3] the weight µis
regarded as a probability measure on [a,b]and the inequalities are interpreted in terms
of the convex ordering between random variables. Recently, also in [10–13, 15, 16] are
studied the Hermite-Hadamard inequalities based on the con vex ordering properties.
Rajba [ 12] was the first who used the Ohlin’s lemma [ 9] on convex stochastic ordering,
to get a simple proof of some known Hermite-Hadamard type ine qualities as well as to
obtaining new Hermite-Hadamard type inequalities. In [ 12] are given some measures
µsatisfying the inequality ( 4), as well as some generalizations of the inequality ( 3).
In this context, we will prove that the majorant and the minor ant in inequalities of
type ( 4) can be respectively obtained by a continuous deformation o f given probability
measure µ. We think this preoccupation is instructive, helping to bet ter understand this
inequality.
Let us recall some basic notions and results on the stochasti c convex order (see,
for example, [ 14]).
DEFINITION 1. Let ξandηbe two random variables. We say that
1.ξis smaller than ηin the convex order (denoted by ξ/lessorequalslantcxη) if
E[f(ξ)]/lessorequalslantE[f(η)]
for all real convex functions f.
2.ξis smaller than ηin the increasing convex order (denoted by ξ/lessorequalslanticxη) if
E[f(ξ)]/lessorequalslantE[f(η)]
for all increasing real convex functions f.
The following theorem (see Theorem 3.A.1 in [ 14]) establishes an useful criterion
for the convex order and the increasing convex order.

CONVEX ORDERING PROPERTIES AND APPLICATIONS 1247
THEOREM 5.LetF and G be the survival functions of the random variables ξ
and η, respectively, that is F(t) =P{ξ>t}and G(t) =P{η>t}, for t ∈R. If
E[ξ] =E[η]then
ξ/lessorequalslantcxη⇔/integraldisplay∞
xF(u)du/lessorequalslant/integraldisplay∞
xG(u)du,∀x∈R⇔ξ/lessorequalslanticxη.
For a random variable ξ, with values in [a,b], let us denote:
•F(x) =P{ξ/lessorequalslantx},x∈R,–the distribution function ofξ;
•p:[a,b]→[0,∞)–the density function ofξ(if exists!);
•E[f(ξ)] =/integraltext
Rf(x)dF(x) =/integraltextb
af(x)p(x)dx–the mean (orthe expectation ) of the
random variable f(ξ)(where f:[a,b]→Ris integrable).
In [3] the inequality ( 1) is written as
f(E[ξ])/lessorequalslantE[f(ξ)]/lessorequalslantE[f(ξ∗)], (5)
where ξis a random variable with uniform distribution on [a,b],ξ∗a random variable
with uniform distribution on {a,b}and f:[a,b]→Ris a convex function. In fact, the
right inequality of ( 5) say
ξ/lessorequalslantcxξ∗.
The same authors provide the following general multi-dimen sional extension.
THEOREM 6.Let K ⊂Rnbe a compact convex set. Denote K∗the set extreme
points of K . For a given K -valued random vector ξ, there is a K∗-valued random
vector ξ∗such that the multi-dimensional Hermite-Hadamard type ine quality
E[f(ξ)]/lessorequalslantE[f(ξ∗)] (6)
holds for all convex functions f :K→R.
In this paper, we will develop the treatment of the weighted H ermite-Hadamard
inequalities in the frame of convex order.
2. Main results
In the following, we will consider a real interval [a,b]and a fixed point m∈(a,b).
Let us define the discrete random variables ξ0andξ1with values in [a,b], having the
distributions
ξ0:/parenleftbigg
m
1/parenrightbigg
and ξ1:/parenleftbigga b
b−m
b−am−a
b−a/parenrightbigg
. (7)
The corresponding survival functions of ξ0andξ1are
F0(x) =/braceleftbigg
1,x<m
0,x/greaterorequalslantmandF1(x) =

1,x<a
m−a
b−a,a/lessorequalslantx<b
0,x/greaterorequalslantb, (8)

1248 A. F LOREA , E. P ˘ALT˘ANEA AND D. B ˘AL˘A
respectively. For an integrable function f:[a,b]→R, we have
E[f(ξ0)] = f(m)andE[f(ξ1)] =b−m
b−af(a)+m−a
b−af(b). (9)
Also, note that the two random variables have the same mean mand
m=a+/integraldisplayb
aF0(x)dx=a+/integraldisplayb
aF1(x)dx.
In fact, we have a specific formula for the mean of [a,b]-valued random variables.
LEMMA 1.Letξbe a random variable taking the values in the interval [a,b]. If
F is the survival function of ξ, then
E[ξ] =a+/integraldisplayb
aF(x)dx.
Proof. LetF=1−F,F(x) =P{ξ/lessorequalslantx},x∈R, be the distribution function of ξ.
Recall that Fis right-continuous. Since F(x) =0 for x<aandF(x) =1 for x/greaterorequalslantb,
we obtain/integraltexta
−∞xdF(x) =a(F(a)−0) =aF(a)and/integraltext∞
bxdF(x) =0. Hence
E[ξ] =/integraldisplay∞
−∞xdF(x) =/integraldisplaya
−∞xdF(x)+/integraldisplayb
axdF(x)+/integraldisplay∞
bxdF(x) =aF(a)−/integraldisplayb
axdF(x).
Integrating by parts, we find
/integraldisplayb
axdF(x) =bF(b)−aF(a)−/integraldisplayb
aF(x)dx=−/bracketleftbigg
aF(a)+/integraldisplayb
aF(x)dx/bracketrightbigg
.
Since F(a)+F(a) =1, we obtain the conclusion. /square
Now, we can adapt Theorem 5to the case of random variables with values in a
given interval.
LEMMA 2.Letξand ηtwo random variables taking values in [a,b], with the
survival functions F and G, respectively. Assume/integraltextb
aF(x)dx=/integraltextb
aG(x)dx. Then ξ/lessorequalslantcx
ηif and only if/integraltextb
tF(x)dx/lessorequalslant/integraltextb
tG(x)dx, for all t ∈(a,b).
Proof. According to our hypothesis and Lemma 1, we find E[ξ] =E[η]. On the
other hand, we have
/integraltext∞
tF(x)dx=a−t+/integraltextb
aF(x)dx=a−t+/integraltextb
aG(x)dx=/integraltext∞
tG(x)dx,fort<a;
/integraltext∞
tF(x)dx=/integraltextb
tF(x)dx/lessorequalslant/integraltextb
tG(x)dx=/integraltext∞
tG(x)dx, fora/lessorequalslantt<b
/integraltext∞
tF(x)dx=0=/integraltext∞
tG(x)dx, fort/greaterorequalslantb
Then, applying Theorem 5, we obtain the conclusion. /square
A significant consequence of the above lemmas is the followin g probabilistic ver-
sion of Theorem 4.

CONVEX ORDERING PROPERTIES AND APPLICATIONS 1249
THEOREM 7.For a random variable ξtaking values in [a,b], with the mean
E[ξ] =m, we have
ξ0/lessorequalslantcxξ/lessorequalslantcxξ1.
In particular, if ξhas a density function p :[a,b]→[0,∞), then
f(m)/lessorequalslant/integraldisplayb
af(x)p(x)dx/lessorequalslantb−m
b−af(a)+m−a
b−af(b), (10)
for any convex function f :[a,b]→R.
Proof. Based on Lemma 1and Lemma 2, it suffices to show
/integraldisplayb
tF0(x)dx/lessorequalslant/integraldisplayb
tF(x)dx/lessorequalslant/integraldisplayb
tF1(x)dx,∀t∈[a,b].
Fort∈[m,b], we have/integraltextb
tF(x)dx/greaterorequalslant0=/integraltextb
tF0dx. Assume that there is t∈[a,m]such
that/integraltextb
tF(x)</integraltextb
tF0(x)dx=m−t. In this case,/integraltextb
aF(x)dx</integraltextt
aF(x)dx+m−t/lessorequalslant
t−a+m−t=m−a, in contradiction with the assumption E[ξ] =m. So/integraltextb
tF0(x)dx/lessorequalslant/integraltextb
tF(x)dx,∀t∈[a,b]. Assume now that there is t∈(a,b)such that/integraltextb
tF(x)dx>/integraltextb
tF1(x)dx=(m−a)(b−t)
b−a. Since the function Fis nonincreasing, we have/integraltextb
tF(x)dx/lessorequalslant
(b−t)F(t). Thus, we obtain F(t)>m−a
b−a. Therefore,/integraltextb
aF(x)dx=/integraltextt
aF(x)dx+
/integraltextb
tF(x)dx/greaterorequalslant(t−a)F(t)+/integraltextb
tF(x)dx>(m−a)(t−a)
b−a+(m−a)(b−t)
b−a=m−a, in contradic-
tion with the assumption E[ξ] =m. So/integraltextb
tF(x)dx/lessorequalslant/integraltextb
tF1(x)dx,∀t∈[a,b]. There-
fore, ξ0/lessorequalslantcxξ/lessorequalslantcxξ1.That is E[f(ξ0)]/lessorequalslantE[f(ξ)]/lessorequalslantE[f(ξ1)], for a convex function
f:[a,b]→R. But, if ξhas the density pthenE[f(ξ)] =/integraltextb
af(x)p(x)dxand we
obtain ( 10). This completes the proof. /square
REMARK 1. Using the Ohlin’s lemma [ 9], Rajba [ 11] gave an alternative very
simple proof of Theorem 7. The Ohlin’s lemma is also used to st udy inequalities of the
Hermite-Hadamard type in the papers [10,13,15,16]. In the p apers [10,15,16], further-
more, to examine the Hermite-Hadamard type inequalities is used the Levin-Steˇ ckin’s
theorem [ 7] (see also [ 8]), as well as Lemma 2 [ 10], which is some modification of the
Levin-Steˇ ckin’s theorem. The Levin-Steˇ ckin’s theorem [ 7] as well as Lemma 2 in [ 10]
give necessary and sufficient conditions for the stochastic convex ordering. Moreover,
for random variables with values in some closed interval, fr om the Levin-Steˇ ckin’s the-
orem [ 7] it follows Theorem 5.
Let us consider now a (fixed) continuous random variable ξtaking values in [a,b].
Assume that ξhas a density function p:[a,b]→[0,∞)(i.e. pis integrable on [a,b],
with/integraltextb
ap(x)dx=1) and the mean E[ξ] =/integraltextb
axp(x)dx=m. Our goal is to show that
the interval between the Hermite-Hadamard bounds

H-H minorant/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
E[f(ξ0)]···/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
left spacingE[f(ξ)]···/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
right spacingH-H majorant/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright
E[f(ξ1)]
.

1250 A. F LOREA , E. P ˘ALT˘ANEA AND D. B ˘AL˘A
can be “filled” by considering an associated family (ξλ)λ∈[0,1]of random variables
which is totally ordered with respect to the stochastic conv ex order.
LetF(x) =P{ξ/lessorequalslantx},x∈Rbe the distribution function of ξ. For x∈[a,b],
we have F(x) =/integraltextx
ap(t)dt. Since Fis a continuous (nondecreasing) function, with
F(a) =0 and F(b) =1, there is c∈(a,b)such that
F(c) =/integraldisplayc
ap(t)dt=b−m
b−a∈(0,1). (11)
DEFINITION 2. Let p:[a,b]→[0,∞)be a density function with the mean m∈
(a,b)andc∈(a,b)such that the relation ( 11) holds. The parametric family of func-
tions(pλ)λ∈(0,1), where pλ:[a,b]→[0,∞), induced by the density p is defined as
follows:
pλ(x) =

1
2(1−λ)p/parenleftBig
x−a(2λ−1)
2(1−λ)/parenrightBig
,x∈[a,uλ]
0, x∈(uλ,vλ)
1
2(1−λ)p/parenleftBig
x−b(2λ−1)
2(1−λ)/parenrightBig
,x∈[vλ,b],forλ∈[1/2,1), (12)
where /braceleftBigg
uλ:= (2λ−1)a+2(1−λ)c
vλ:= (2λ−1)b+2(1−λ)c,
and
pλ(x) =/braceleftBigg1
2λp/parenleftBig
x+m(2λ−1)
2λ/parenrightBig
,x∈[sλ,tλ]
0, x∈[a,sλ)∪(tλ,b],forλ∈(0,1/2), (13)
where /braceleftBigg
sλ:=2λa+(1−2λ)m
tλ:=2λb+(1−2λ)m.
Observe that p1/2=p. Moreover, the definition ( 13) can be even applied for λ=
1/2. The significance of the family of functions (pλ)λ∈(0,1)introduced in Definition 2
is highlighted by the following lemma.
LEMMA 3.pλ:[a,b]→[0,∞)is a density function on [a,b]with the mean m, for
allλ∈(0,1).
Proof. Clearly, pλis integrable on [a,b]forλ∈(0,1).
Assume λ∈(1/2,1). We have
/integraldisplayuλ
apλ(x)dx=1
2(1−λ)/integraldisplay(2λ−1)a+2(1−λ)c
ap/parenleftbiggx−a(2λ−1)
2(1−λ)/parenrightbigg
dx=/integraldisplayc
ap(t)dt
and
/integraldisplayb
vλpλ(x)dx=1
2(1−λ)/integraldisplayb
(2λ−1)b+2(1−λ)cp/parenleftbiggx−b(2λ−1)
2(1−λ)/parenrightbigg
dx=/integraldisplayb
cp(t)dt.

CONVEX ORDERING PROPERTIES AND APPLICATIONS 1251
Then
/integraldisplayb
apλ(x)dx=/integraldisplayuλ
apλ(x)dx+/integraldisplayb
vλpλ(x)dx=/integraldisplayc
ap(t)dt+/integraldisplayb
cp(t)dt=/integraldisplayb
ap(t)dt=1.
In addition, from ( 11), we obtain
/integraldisplayuλ
apλ(x)dx=b−m
b−aand/integraldisplayb
vλpλ(x)dx=m−a
b−a. (14)
The mean of pλis given by
/integraldisplayb
axpλ(x)dx=1
2(1−λ)/integraldisplayuλ
axp/parenleftbiggx−a(2λ−1)
2(1−λ)/parenrightbigg
dx+1
2(1−λ)/integraldisplayb
vλxp/parenleftbiggx−b(2λ−1)
2(1−λ)/parenrightbigg
dx
=/integraldisplayc
ap(t)[2(1−λ)t+a(2λ−1)]dt+/integraldisplayb
cp(t)[2(1−λ)t+b(2λ−1)]dt
=2(1−λ)/integraldisplayb
at p(t)dt+(2λ−1)/parenleftbigg
a/integraldisplayc
ap(t)dt+b/integraldisplayb
cp(t)dt/parenrightbigg
=2(1−λ)m+(2λ−1)/bracketleftbigga(b−m)
b−a+b(m−a)
b−a/bracketrightbigg
=m.
Forλ∈(0,1/2), we find
/integraldisplayb
apλ(x)dx=1
2λ/integraldisplay2λb+(1−2λ)m
2λa+(1−2λ)mp/parenleftbiggx+m(2λ−1)
2λ/parenrightbigg
dx=/integraldisplayb
ap(t)dt=1
and
/integraldisplayb
axpλ(x)dx=/integraldisplayb
ap(t)[2λt+m(1−2λ)]dt=2λ/integraldisplayb
at p(t)dt+m(1−2λ)/integraldisplayb
ap(t)dt=m.
Therefore the pλ:[a,b]→[0,∞)is a density function on [a,b]with the mean m, for
allλ∈(0,1)./square
Based on the above construction of the family of density func tions(pλ)λ∈(0,1)
(Definition 2), we will formulate the main result of this paper.
THEOREM 8.Letξλbe a continuous random variable with the density function
pλ, for λ∈(0,1). Let ξ0and ξ1be two random variable with the distribution de-
fined by ( 7). The family (ξλ)λ∈[0,1]of[a,b]-valued random variables has the following
properties:
1.ξλ(d)→ξ1, for λ↑1;
2.ξλ(d)→ξ0, for λ↓0;
3.ξλ(d)→ξλ0, for λ→λ0;

1252 A. F LOREA , E. P ˘ALT˘ANEA AND D. B ˘AL˘A
4. if λ,µ∈[0,1], such that λ<µ, then ξλ/lessorequalslantcxξµ;
5. for every convex function f :[a,b]→R,
If:[0,1]→R,If(λ):=E[f(ξλ)] =/integraldisplayb
af(x)pλ(x)dx,
is a continuous nondecreasing function on [0,1], with the image
If([0,1]) =/bracketleftbigg
f(m),(b−m)f(a)+(m−a)f(b)
b−a/bracketrightbigg
.
Here, “(d)→” denotes the convergence in distribution.
Proof. Forλ∈[0,1], let us denote by FλandFλthe distribution function and
the survival function of ξλ, respectively.
1. We have to show lim
λ→1−Fλ(x) =F1(x)at the continuity points x∈[a,b]of
F1. Assume x∈(a,b). Since lim
λ→1−uλ=aand lim
λ→1−vλ=bthere is λx∈(1/2,1)
such that uλ<x<vλ,∀λ∈(λx,1). Then, from ( 14),Fλ(x) =/integraltextuλapλ(t)dt=b−m
b−a=
F1(x),∀λ∈(λx,1). So, ξλ(d)→ξ1, for λ↑1.
2. For x∈[a,m), there is λx∈(0,1/2)such that sλ>x,∀λ∈(0,λx). Then
Fλ(x) =0=F0(x),∀λ∈(0,λx). Similarly, for x∈(m,b], there is λx∈(0,1/2)such
thattλ<x,∀λ∈(0,λx). Thus, Fλ(x) =/integraltexttλsλpλ(t)dt=1=F0(x),∀λ∈(0,λx). As
follows, ξλ(d)→ξ0, for λ↓0.
3. It is sufficient to note that the function Fλ(x) =/integraltextx
apλ(t)dtis continuous in
λ∈(0,1), for all x∈[a,b].
4. Theorem 7ensures ξ0/lessorequalslantcxξλ/lessorequalslantcxξ1, for all λ∈(0,1). Then, we have only
to establish the convex ordering between ξλand ξµfor 0<λ<µ/lessorequalslant1/2 and for
1/2/lessorequalslantλ<µ<1. Let ϕ:[a,b]→Rbe the function defined by
ϕ(x) =/integraldisplayb
xFµ(t)dt−/integraldisplayb
xFλ(t)dt,x∈[a,b].
We have ϕ′(x) =Fλ(x)−Fµ(x) =Fµ(x)−Fλ(x),forx∈[a,b].
In the first case, 0 <λ<µ/lessorequalslant1/2, we obtain
Fλ(x) =/integraldisplayx
apλ(t)dt=

0, x∈[a,sλ]
/integraltextm+x−m
2λa p(z)dz,x∈(sλ,tλ)
1, x∈[tλ,b].

CONVEX ORDERING PROPERTIES AND APPLICATIONS 1253
Thus, after some calculations, we find
ϕ′(x) =

0, x∈/bracketleftbig
a,sµ/bracketrightbig
/integraltextm+x−m
2µ
a p(z)dz,x∈/parenleftbig
sµ,sλ/parenrightbig
/integraltextm+x−m
2µ
m+x−m
2λp(z)dz,x∈[sλ,m]
−/integraltextm+x−m
2λ
m+x−m
2µp(z)dz,x∈(m,tλ)
−/integraltextb
m+x−m
2µp(z)dz,x∈/bracketleftbig
tλ,tµ/bracketrightbig
0, x∈/parenleftbig
tµ,b/bracketrightbig.
Therefore, according to the sign of the derivative, ϕis nondecreasing on the interval
[0,m]and nonincreasing on the interval [m,b]. Since ϕ(a) = ( m−a)−(m−a) =0
andϕ(b) =0−0=0, we conclude that ϕ(x)/greaterorequalslant0,∀x∈[a,b].
In the second case, 1 /2/lessorequalslantλ<µ<1, we have
Fλ(x) =/integraldisplayx
apλ(t)dt=

/integraltexta+x−a
2(1−λ)
a p(z)dz, x∈[a,uλ]
b−m
b−a, x∈(uλ,vλ)
1−/integraltextb
b−b−x
2(1−λ)p(z)dz,x∈[vλ,b]
and
ϕ′(x) =

/integraltexta+x−a
2(1−µ)
a+x−a
2(1−λ)p(z)dz,x∈/bracketleftbig
a,uµ/bracketrightbig
/integraltextc
a+x−a
2(1−λ)p(z)dz,x∈/parenleftbig
uµ,uλ/parenrightbig
0, x∈[uλ,vλ]
−/integraltextb−b−x
2(1−λ)
c p(z)dz,x∈/parenleftbig
vλ,vµ/parenrightbig
−/integraltextb−b−x
2(1−λ)
b−b−x
2(1−µ)p(z)dz,x∈/bracketleftbig
vµ,b/bracketrightbig.
Hence, ϕis nondecreasing on the interval [0,vλ]and nonincreasing on the interval
[vλ,b]. From ϕ(a) =ϕ(b) =0, we conclude that ϕ(x)/greaterorequalslant0,∀x∈[a,b].
As follows, for 0 <λ<µ<1, we have/integraltextb
xFλ(t)dt/lessorequalslant/integraltextb
xFµ(t)dt,∀x∈[a,b],
with/integraltextb
aFλ(t)dt=/integraltextb
aFµ(t)dt. Thus, from Lemma 2, we obtain ξλ/lessorequalslantcxξµ, for 0 <
λ<µ<1.
5. Let f:[a,b]→Rbe a convex function. From the definition of the convex order
and the previous result 4, the function Ifis nondecreasing on [0,1]. We have
If(λ) =/integraldisplaytλ
sλf(x)pλ(x)dx=/integraldisplayb
af(m+2λ(t−m))p(t)dt,forλ∈(0,1/2) (15)

1254 A. F LOREA , E. P ˘ALT˘ANEA AND D. B ˘AL˘A
and
If(λ) =/integraldisplayuλ
af(x)pλ(x)dx+/integraldisplayb
vλf(x)pλ(x)dx (16)
=/integraldisplayc
af(a+2(1−λ)(t−a))p(t)dt
+/integraldisplayb
cf(b+2(1−λ)(t−b))p(t)dt,forλ∈[1/2,1).
The function fis then bounded on [a,b]and continuous on (a,b). Clearly, we have
If(λ) =If(λ)where fis the continuous function on [a,b]defined by f(x) =f(x),for
x∈(a,b),f(a) = lim
x→a+f(x)and f(b) = lim
x→b−f(x). Then, based on the properties of
the integrals involving parameters, the function Ifis continuous on (0,1). Also, we can
directly verify lim
λ→0+If(λ) =If(0)and lim
λ→1−If(λ) =If(1). Hence Ifis continuous
on[0,1], with the image/bracketleftBig
f(m),(b−m)f(a)+(m−a)f(b)
b−a/bracketrightBig
./square
REMARK 2. We have If(1/2) =/integraltextb
af(x)p(x)dx.If(0) =f(m)is the Hermite-
Hadamard minorant andIf(1)=b−m
b−af(a)+m−a
b−af(b)is the Hermite-Hadamard majo-
rant. Note that the continuity of Ifat 0 and 1 is explained by the proved convergences
ξλ(d)→ξ0, for λ↓0, and ξλ(d)→ξ1, for λ↑1.
Further, we intend to complete our probabilistic approach b y indicating a general
method to construct a random variable with the values in the i nterval [a,b]and the
expectation m∈(a,b). The construction starts from an arbitrary [0,1]-valued random
variable. Moreover, starting from a family of [0,1]-valued random variables, which
is totally ordered with respect to the increasing convex ord er, we will get a family of
[a,b]-valued random variables which is totally ordered with resp ect to the convex order.
DEFINITION 3. Assume a real interval [a,b]andm∈(a,b). Let ξbe a random
variable taking the values in the interval [0,1]. Suppose that the distribution function
Fofξhas at most a finite number of points of discontinuity. We defin e the random
variables ξ−=m+ (a−m)ξ, with the distribution function F−, and ξ+=m+ (b−
m)ξ, with the distribution function F+. A random variable/tildewideξhaving the distribution
function F∗=b−m
b−aFξ−+m−a
b−aFξ+, will be called an ([a,b];m)-mixture of ξ.
We easily observe that/tildewideξtakes the values in the interval [a,b]. In particular, /tildewide0=ξ0
and/tildewide1=ξ1(see Definition 7). The following theorem proves some properties of the
([a,b];m)-mixtures.
THEOREM 9.Letξandηbe two random variables satisfying Definition 3, such
that ξ/lessorequalslanticxη.Let us consider their corresponding ([a,b];m)-mixtures/tildewideξand/tildewideη, re-
spectively. Then E/bracketleftBig/tildewideξ/bracketrightBig
=m and/tildewideξ/lessorequalslantcx/tildewideη.

CONVEX ORDERING PROPERTIES AND APPLICATIONS 1255
Proof. LetFandGbe the distribution functions of ξandη, respectively. From
Lemma 1and considering Definition 3we obtain
E/bracketleftBig/tildewideξ/bracketrightBig
=a+/integraldisplayb
aF∗(x)dx=b−m
b−a/bracketleftbigg
a+/integraldisplayb
aF−(x)dx/bracketrightbigg
+m−a
b−a/bracketleftbigg
a+/integraldisplayb
aF+(x)dx/bracketrightbigg
=b−m
b−aE[ξ−]+m−a
b−aE[ξ+]
=b−m
b−a{m+(a−m)E[ξ]}+m−a
b−a{m+(b−m)E[ξ]}=m.
We also have E[/tildewideη] =m. As follows,
/integraldisplayb
aF∗(t)dt=/integraldisplayb
aG∗(t)dt. (17)
The inequality ξ/lessorequalslanticxηimplies
/integraldisplay1
xF(t)dt/lessorequalslant/integraldisplay1
xG(t)dt,∀x∈[0,1]. (18)
From Definition 3F, we obtain
F−(x) =P{ξ−/lessorequalslantx}=P/braceleftbigg
ξ/greaterorequalslantx−m
a−m/bracerightbigg
andF+(x) =P/braceleftbigg
ξ/lessorequalslantx−m
b−m/bracerightbigg
,forx∈[a,b].
Then
F∗(x) =1−/bracketleftbiggb−m
b−aP/braceleftbigg
ξ/greaterorequalslantx−m
a−m/bracerightbigg
+m−a
b−aP/braceleftbigg
ξ/lessorequalslantx−m
b−m/bracerightbigg/bracketrightbigg
=

1−b−m
b−aP/braceleftbig
ξ/greaterorequalslantx−m
a−m/bracerightbig
,forx∈[a,m)
m−a
b−aF/parenleftbigx−m
b−m/parenrightbig
, forx∈(m,b]
Since Fhas at most a finite number of discontinuity points, we obtain after some
calculations
/integraldisplayb
xF∗(t)dt=(b−m)(m−a)
b−a/integraldisplay1
x−m
b−mF(z)dz,forx∈[m,b],
and/integraldisplayb
xF∗(t)dt=(b−m)(m−a)
b−a/integraldisplay1
x−m
a−mF(z)dz+m−x,forx∈[a,m).
But we have similar results for the expression of/integraltextb
xG∗(t)dt. Therefore, using the
relation ( 18), we find
/integraldisplayb
xF∗(t)dt/lessorequalslant/integraldisplayb
xG∗(t)dt,∀x∈[a,b]. (19)
From ( 17), (19) and Lemma 2we deduce/tildewideξ/lessorequalslantcx/tildewideη./square
Finally, we apply our last construction to obtain new result s of Hermite-Hadamard
type.

1256 A. F LOREA , E. P ˘ALT˘ANEA AND D. B ˘AL˘A
DEFINITION 4. A stochastic process (Xλ)λ∈[0,1]with values in [0,1]is said to be
continuous increasing convex ordered if
1.X0=0 and Xλ(d)→X0, for λ↓0;
2.X1=1 and Xλ(d)→X1, for λ↑1;
3.Xλ(d)→Xλ0, for λ→λ0∈(0,1);
4. 0/lessorequalslantλ1<λ2/lessorequalslant1⇒Xλ1/lessorequalslanticxXλ2.
EXAMPLE 1. Let Xλbe a Beta random variable with the distribution function
Fλ(x) =sin(λ π)
π/integraldisplayx
0tλ−1(1−t)−λdt,x∈[0,1],
for all λ∈(0,1). Assume X0=0 and X1=1. Then (Xλ)λ∈[0,1]is a continuous
increasing convex ordered stochastic process.
We obtain an alternative of Theorem 8.
THEOREM 10. Let(Xλ)λ∈[0,1]be a continuous increasing convex ordered stochas-
tic process. For λ∈[0,1], denote /tildewideXλthe([a,b];m)-mixture of X λ. Then
0/lessorequalslantλ1<λ2/lessorequalslant1⇒E/bracketleftBig
f/parenleftBig
/tildewideXλ1/parenrightBig/bracketrightBig
/lessorequalslantE/bracketleftBig
f/parenleftBig
/tildewideXλ2/parenrightBig/bracketrightBig
,
for all convex functions f :[a,b]→R, i.e.(/tildewideXλ)λ∈[0,1]is a totally ordered convex family
of random variables taking values in [a,b], with the common mean m.
The proof is based on Theorem 9. The details are ommited.
Acknowledgements. We express our many thanks to the editor and to the referee
for careful reading the manuscript and for valuable comment s and suggestions, helping
to improve the presentation of this paper.
This work was partially supported by the Grant number 3C/201 4, awarded in the
internal Grant competition of the University of Craiova.
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(Received October 26, 2014) Aurelia Florea
University of Craiova
Romania
e-mail:aurelia florea@yahoo.com
Eugen P˘ alt˘ anea
Transilvania University of Bras ¸ov
Romania
e-mail:epaltanea@unitbv.ro
Dumitru B˘ al˘ a
University of Craiova
Romania
e-mail:dumitru bala@yahoo.com
Journal of Mathematical Inequalities
www.ele-math.com
jmi@ele-math.com