On some recent applications of stochastic convex ordering theorems to some functional inequalities for convex functions - a survey

0.32 MB·

by  Teresa Rajba

On some recent applications of stochastic convex
ordering theorems to some functional
inequalities for convex functions - a survey
TeresaRajba
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a
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Abstract This is a survey paper concerning some theorems on stochastic convex

orderingandtheir applicationsto functionalinequalitiesforconvexfunctions.We
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A presenttherecentresultsonthosesubjects.
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. Keywords: convexfunctions,higherorderconvexfunctions,Hermite-Hadamard
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inequalities,convexstochasticordering.
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MathematicsSubjectClassification(2010)26A51,26D10,39B62.
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1 Introduction
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Inthepresentpaperwe lookatHermite-Hadamardtypeinequalitiesfromthe per-
0
spective provided by the stochastic convex order. This approach is mainly due to
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0 Cal and Cárcamo. In the paper [10], the Hermite-Hadamard type inequalities are
. interpreted in terms of the convex stochastic ordering between random variables.
1
Recently, also in [19, 32, 35, 36, 37, 38, 40, 41, 42]), the Hermite-Hadamard in-
0
7 equalities are studied based on the convex ordering properties. Here we want to
1 attractthereadersattentiontosomeselectedtopicsbypresentingsometheoremson
v: theconvexorderingthatcanbeusefulinthestudyoftheHermite-Hadamardtype
i inequalities.
X
The Ohlin lemma [31] on sufficient conditions for convex stochastic ordering
r
was first used in [36], to get a simple proof of some known Hermite-Hadamard
a
typeinequalitiesaswellasto obtainingnew Hermite-Hadamardtypeinequalities.
In[32,41,42],theauthorsusedtheLevin-Stecˇkintheorem[25]tostudyHermite-
Hadamardtypeinequalities.
TeresaRajba
University of Bielsko-Biala, Willowa 2, 43-309 Bielsko-Biala, Poland, e-mail:
[email protected]
1
2 TeresaRajba
ManyresultsonhigherordergeneralizationsoftheHermite-Hadamardtypein-
equalityonecanfound,amongothers,in[1,2,3,4,5,14,36,37].Inrecentpapers
[36, 37] the theorem of M. Denuit, C.Lefèvre and M. Shaked [13] was used to
proveHermite-Hadamardtypeinequalitiesfor higher-orderconvexfunctions.The
theorem of M. Denuit, C.Lefèvreand M. Shaked [13] on sufficient conditionsfor
s-convexorderingisacounterpartoftheOhlinlemmaconcerningconvexordering.
Atheoremonnecessaryandsufficientconditionsforhigherorderconvexstochas-
tic ordering,which is a counterpartof the Levin-Stecˇkintheorem [25] concerning
convexstochasticordering,isgiveninthepaper[38].Basedonthistheorem,use-
fulcriteriafortheverificationofhigherorderconvexstochasticorderingaregiven.
ThesecriteriacanbeusefulinthestudyofHermite-Hadamardtypeinequalitiesfor
higherorderconvexfunctions,andinparticularinequalitiesbetweenthequadrature
operators.Theymaybeeasiertoverifythehigherorderconvexorders,thanthose
givenin[13,22].
InSection2,wegivesimpleproofsofknownaswellasnewHermite-Hadamard
typeinequalities,usingOhlin’sLemmaandtheLevin-Stecˇkintheorem.
In Sections 3 and 4, we study inequalities of the Hermite-Hadamard type in-
volving numericaldifferentiationformulasof the first order and the second order,
respectively.
InSection5, wegivesimple proofsofHermite-Hadamardtypeinequalitiesfor
higher-orderconvexfunctions,usingthetheoremofM. Denuit,C.LefèvreandM.
Shaked,andageneralizationoftheLevin-Stecˇkintheoremtohigherorders.These
resultsareappliedtoderivesomeinequalitiesbetweenquadratureoperators.
2 Some generalizations ofthe Hermite-Hadamardinequality
Let f: [a,b] R be a convex function (a,b R, a<b). The following double
→ ∈
inequality
a+b 1 b f(a)+f(b)
f f(x)dx (1)
(cid:18) 2 (cid:19)≤ b aZa ≤ 2
−
is known as the Hermite-Hadamardinequality (see [14] for many generalizations
andapplicationsof(1)).
Inmanypapers,theHermite-Hadamardtypeinequalitiesarestudiedbasedonthe
convexstochasticorderingproperties(see,forexample,[19,32,35,36,37,40,41]).
In the paper [36], the Ohlin lemma on sufficient conditions for convex stochastic
orderingisusedtogetasimpleproofofsomeknownHermite-Hadamardtypein-
equalitiesaswellas to obtainnewHermite-Hadamardtypeinequalities.Recently,
the Ohlin lemma is also used to study the inequalities of the Hermite-Hadamard
type for convex functions in [32, 35, 40, 41]. In [37], also the inequalities of the
Hermite-Hadamardtypefordelta-convexfunctionsare studiedbyusingthe Ohlin
lemma.Inthepapers[32,40,41],furthermore,theLevin-Stecˇkintheorem[25](see
also[30])isusedtoexaminetheHermite-Hadamardtypeinequalities.Thistheorem
givesnecessaryandsufficientconditionsforthestochasticconvexordering.
Stochasticconvexorderingtheoremstofunctionalinequalities 3
Letusrecallsomebasicnotionsandresultsonthestochasticconvexorder(see,
forexample,[13]).Asusual,F denotesthedistributionfunctionofarandomvari-
X
ableX andm isthedistributioncorrespondingtoX.Forrealvaluedrandomvari-
X
ables X,Y with a finite expectation, we say that X is dominated by Y in convex
orderingsense,if
Ef(X) Ef(Y)
≤
forallconvexfunctions f: R R (forwhich theexpectationsexist). Inthatcase
wewriteX Y,orm m →.
cx X cx Y
≤ ≤
Inthe followingOhlin’slemma[31], aregivensufficientconditionsforconvex
stochasticordering.
Lemma1(Ohlin [31]). Let X,Y be two randomvariables such that EX =EY. If
thedistributionfunctionsF ,F crossexactlyonetime,i.e.,forsomex holds
X Y 0
F (x) F (x)ifx<x and F (x) F (x)ifx>x ,
X Y 0 X Y 0
≤ ≥
then
Ef(X) Ef(Y) (2)
≤
forallconvexfunctions f: R R.
→
The inequality (1) may be easily proved with the use of the Ohlin lemma
(see[36]). Indeed, let X, Y, Z be three random variables with the distributions
m =d ,m whichisequallydistributedin[a,b]andm = 1(d +d ),respec-
X (a+b)/2 Y Z 2 a b
tively.Thenitiseasytoseethatthepairs(X,Y)and(Y,Z)satisfytheassumptions
oftheOhlinlemma,andusing(2),weobtain(1).
Leta<c<d<b.Let f: I Rbeaconvexfunction,a,b I.Then(see[21])
→ ∈
f(c)+f(d) c+d f(a)+f(b) a+b
f f . (3)
2 − (cid:18) 2 (cid:19)≤ 2 − (cid:18) 2 (cid:19)
Toprove(3)fromtheOhlinlemma,itsufficestotakerandomvariablesX,Y (see
[27])with
1 1
m = (d +d )+ d ,
X 4 c d 2 (a+b)/2
1 1
m Y = (d a+d b)+ d d .
4 2 (c+d)/2
Then,byLemma1,weobtain
f(c)+f(d) a+b f(a)+f(b) c+d
+f +f , (4)
2 (cid:18) 2 (cid:19)≤ 2 (cid:18) 2 (cid:19)
whichimplies(3).
4 TeresaRajba
Similarly,itcanbeprovedthePopoviciuinequality
2 x+y y+z z+x f(x)+f(y)+f(z) x+y+z
f +f +f +f ,
3(cid:20) (cid:18) 2 (cid:19) (cid:18) 2 (cid:19) (cid:18) 2 (cid:19)(cid:21)≤ 3 (cid:18) 3 (cid:19)
(5)
where x,y,z I and f: I R is a convex function. To prove (5) from the Ohlin
∈ →
lemma, it suffices (assuming x y z ) to take random variables X,Y (see [27])
≤ ≤
with
1
m = d +d +d ,
X 4 (x+y)/2 (y+z)/2 (z+x)/2
(cid:0) (cid:1)
1 1
m = (d +d +d )+ d .
Y 6 x y z 2 (x+y+z)/3
Convexityhasaniceprobabilisticcharacterization,knownasJensen’sinequality
(see[6]).
Proposition1([6]).Afunction f: (a,b) Risconvexif,andonlyif,
→
f(EX) Ef(X) (6)
≤
forall(a,b)-valuedintegrablerandomvariablesX.
Toprove(6)fromtheOhlinlemma,itsufficestotakearandomvariableY (see
[35])with
m Y =d EX,
thenwehave
Ef(Y)= f(EX). (7)
By the Ohlin lemma, we obtain Ef(Y) Ef(X), then taking into account(7),
≤
thisimplies(6).
Remark1.Note,thatin[29],theOhlinlemmawasusedtoobtainasolutionofthe
problemofRas¸aconcerninginequalitiesforBernsteinoperators.
In[17],Fejérgaveageneralizationoftheinequality(1).
Proposition2([17]).Let f: I RbeaconvexfunctiondefinedonarealintervalI,
a,b Iwitha<bandletg: [a→,b] Rbenonnegativeandsymmetricwithrespect
∈ →
tothepoint(a+b)/2(theexistenceofintegralsisassumedinallformulas).Then
a+b b b f(a)+f(b) b
f g(x)dx f(x)g(x)dx g(x)dx. (8)
(cid:18) 2 (cid:19)·Za ≤Za ≤ 2 ·Za
Thedoubleinequality(8)isknownintheliteratureastheFejérinequalityorthe
Hermite-Hadamard-Fejérinequality(see[14,28,33]forthehistoricalbackground).
Stochasticconvexorderingtheoremstofunctionalinequalities 5
Remark2([36]).UsingtheOhlinlemma(Lemma1),wegetasimpleproofof(8).
Let f andgsatisfytheassumptionsofProposition2.LetX,Y, Z bethreerandom
variablessuchthatm =d ,m (dx)=( bg(x)dx) 1g(x)dx,m = 1(d +d ).
X (a+b)/2 Y a − Z 2 a b
Then,byLemma1,weobtainthatX cxY anRdY cxZ,whichimplies(8).
≤ ≤
Remark3.Notethatforg(x)=w(x)suchthat bw(x)dx=1,theinequality(8)can
a
berewrittenintheform R
a+b b f(a)+f(b)
f f(x)w(x)dx . (9)
(cid:18) 2 (cid:19)≤Za ≤ 2
Conversely, from the inequality (9), it follows (8). Indeed, if bg(x)dx>0, it
a
sufficestotakew(x)= bg(x)dx −1g(x).If bg(x)dx=0,thenR(8)isobvious.
a a
(cid:16)R (cid:17) R
For various modifications of (1) and (8) see e.g. [3, 4, 5, 11, 12, 14], and the
referencesgiventhere.
As Fink noted in [18], one wonders what the symmetry has to do with the in-
equality(8)andifsuchaninequalityholdsforotherfunctions(cf.[14,p.53]).
As an immediate consequence of Lemma 1, we obtain the following theorem,
whichisageneralizationoftheFejérinequality.
Theorem1 ([36]).Let0< p<1.Let f: I Rbeaconvexfunction,a,b I with
a<b. Let m be a finite measure on B([a,→b]) such that (i) m ([a,pa+qb])∈ pP,
0
(ii) m ((pa+qb,b]) qP,(iii) xm (dx)=(pa+qb)P,whereq=1 p≤,P =
≤ 0 [a,b] 0 − 0
m ([a,b]).Then R
f(pa+qb)P f(x)m (dx) [pf(a)+qf(b)]P. (10)
0 0
≤Z[a,b] ≤
Fink proved in [18] a general weighted version of the Hermite-Hadamard in-
equality.Inparticular,wehavethefollowingprobabilisticversionofthisinequality.
Proposition3([18]).LetX bearandomvariabletakingvaluesintheinterval[a,b]
suchthatm istheexpectationofX and m isthedistributioncorrespondingto X.
X
Then
b b m m a
f(m) f(x)m X(dx) − f(a)+ − f(b). (11)
≤Za ≤ b a b a
− −
Moreover,in[19]itwasprovedthat,startingfromsuchafixedrandomvariable
X, we can fill the whole space between the Hermite-Hadamard bounds by high-
lighting some parametric families of random variables. The authors propose two
alternativeconstructionsbasedontheconvexorderingproperties.
In[35],basedonLemma1,averysimpleproofofProposition3isgiven.LetX
bearandomvariablesatisfyingtheassumptionsofProposition3.LetY,Z betwo
randomvariablessuchthatm =d ,m = b md +m ad ).Then,byLemma1,we
Y m Z b−a a b−a b
obtainthatY X andX Z,whichimp−lies(11).−
cx cx
≤ ≤
6 TeresaRajba
In[36],someresultsrelatedtotheBrenner-Alzerinequalityaregiven.Inthepa-
per[23]byM.Klaricˇic´Bakula,J.Pecˇaric´andJ.Peric´,someimprovementsofvari-
ousformsoftheHermite-Hadamardinequalitycanbefound;namely,thatofFejér,
Lupas, Brenner-Alzer, Beesack-Pecˇaric´. These improvementsimply the Hammer-
Bulleninequality.In1991,BrennerandAlzer[9]obtainedthefollowingresultgen-
eralizingFejér’sresultaswellastheresultofVasic´ andLackovic´ (1976)[43]and
Lupas(1976)[26](seealso[33]).
Proposition4([9]).Let p,qbegivenpositivenumbersanda a<b b .Then
1 1
≤ ≤
theinequalities
pa+qb 1 A+y pf(a)+qf(b)
f f(t)dt (12)
(cid:18) p+q (cid:19)≤ 2yZA y ≤ p+q
−
holdforA= pa+qb,y>0,andallcontinuousconvexfunctions f: [a ,b ] R if,
p+q 1 1 →
andonlyif,
b a
y − min p,q .
≤ p+q { }
Remark4.Itisknown[33,p.144]thatunderthesameconditionsHermite-Hadamard’s
inequalityholds,thefollowingrefinementof(12):
pa+qb 1 A+y 1 pf(a)+qf(b)
f f(t)dt f(A y)+f(A+y)
(cid:18) p+q (cid:19)≤ 2yZA y ≤ 2{ − }≤ p+q
− (13)
holds.
InthefollowingtheoremwegivesomegeneralizationoftheBrennerandAlzer
inequalities(13),whichweproveusingtheOhlinlemma.
Theorem2 ([36] ). Let p,q be given positive numbers, a a<b b , 0<y
1 1
b amin p,q andlet f: [a ,b ] Rbeaconvexfunction.≤Then ≤ ≤
p−+q { } 1 1 →
pa+qb
f
(cid:18) p+q (cid:19)≤
a 1 A+(1 a )y
f(A (1 a )y)+f(A+(1 a )y) + − f(t)dt
2 { − − − } 2yZA (1 a )y ≤
− −
a (cid:229) n f A y+ka y +f A+y ka y + 1 A+(1−a )yf(t)dt
2nk=1n (cid:16) − n (cid:17) (cid:16) − n (cid:17)o 2yZA−(1−a )y ≤
1 A+y
f(t)dt, (14)
2yZA y
−
where0 a 1,n=1,2,...,
≤ ≤
Stochasticconvexorderingtheoremstofunctionalinequalities 7
1 A+y b 1 A+y
f(t)dt f(A y)+f(A+y) +(1 b ) f(t)dt
2yZA y ≤ 2{ − } − 2yZA y
− −
1
f(A y)+f(A+y) , (15)
≤ 2{ − }
where0 b 1,
≤ ≤ 1
f(A y)+f(A+y)
2{ − }≤
1
( g ) f(A y c)+f(A+y+c) +g f(A y)+f(A+y)
2− { − − } { − }≤
pf(a)+qf(b)
, (16)
p+q
wherec=min b (A+y),(A y) a ,g = 1 p .
{ − − − } 2−
(cid:12) (cid:12)
(cid:12) (cid:12)
Toprovethistheorem,itsufficestoconsiderrandomvariablesX,Y,W,Z,x ,h
n
andl suchthat:
m = d ,
X pa+qb
p+q
1
m (dx)= c (x)dx,
Y 2y [A−y,A+y]
p q 1 1
m = d + d ,m = d + d ,
Z a b W A y A+y
p+q p+q 2 − 2
m xn(dx)= 2ank(cid:229)=n1{d A−y+kany +d A+y−kany}+21yc [A−(1−a )y,A+(1−a )y](x)dx,
b 1 b
m h (dx)= 2{d A−y+d A+y}+ 2−y c [A−y,A+y](x)dx,
1
m l = ( g ) d A y c+d A+y+c +g d A y+d A+y .
2− { − − } { − }
Then,usingtheOhlinlemma,weobtain:
X Y,Y W andW Z,whichimpliestheinequalities(13),
cx cx cx
• X ≤ x ,x ≤ x andx ≤ Y,whichimplies(14),
cx 1 1 cx n n cx
• Y ≤ h and≤h W,whic≤himplies(15),
cx cx
• W≤ l andl ≤ Z,whichimplies(16).
cx cx
• ≤ ≤
Theorem3 ([36] ). Let p,q be given positive numbers, 0<a <1, a a<b
1
b ,0<y b amin p,q and0 a y b amin p,q .Let f: [a ,b≤] Rb≤e
1 ≤ p−+q { } ≤ 1 a ≤ p−+q { } 1 1 →
aconvexfunction.Then −
8 TeresaRajba
a A (1 a )2 A+1aa y
f(A) f(t)dt+ − − f(t)dt
≤ y ZA y a y ZA
− a
a f(A y)+(1 a )f(A+ y)
≤ − − 1 a
p q −
f(a)+ f(b), (17)
≤ p+q p+q
whereA= pa+qb.
p+q
LetX,Y,ZandW berandomvariablessuchthat:
m = d ,
X A
a (1 a )2
m Y(dx)= yc [A−y,A](x)dx+ −a y c [A,A+1−aa y](x)dx,
m W = ad A−y+(1−a )d A+1aa y,
p q −
m = d + d .
Z a b
p+q p+q
Then,usingtheOhlinlemma,weobtainX Y,Y W,W Z,whichimplies
cx cx cx
≤ ≤ ≤
theinequalities(17).
Remark5.If we choose a = 1 in Theorem 3, then the inequalities(17) reduce to
2
theinequalities(15).
Remark6.Ifwechoosea = p andy=(1 p)zinTheorem3,thenwehave
p+q −
p A q A+p+pqz
f(A) f(t)dt+ f(t)dt
≤ qzZA q z pzZA
−p+q
p q q p
f(A z)+ f(A+ z)
≤ p+q − p+q p+q p+q
p q
f(a)+ f(b),
≤ p+q p+q
whereA= pa+qb,0<z b a.
p+q ≤ −
In the paper [40], the author used Ohlin’s lemma to prove some new inequali-
tiesoftheHermite-Hadamardtype,whichareageneralizationofknownHermite-
Hadamardtypeinequalities.
Theorem4 ([40]).Theinequality
1 y
af(a x+(1 a )y)+(1 a)f(b x+(1 b )y) f(t)dt, (18)
− − − ≤ y xZx
−
withsomea,a ,b [0,1],a >b issatisfiedforallx,y Randallcontinuousand
convexfunctions f∈:[x,y] Rif,andonlyif, ∈
→
Stochasticconvexorderingtheoremstofunctionalinequalities 9
1
aa +(1 a)b = , (19)
− 2
andoneofthefollowingconditionsholdstrue:
(i)a+a 1,
(ii)a+b ≤ 1,
(iii)a+a ≥>1,a+b <1anda+2a 2.
≤
Theorem5 ([40]).Let a,b,c,a (0,1)be numberssuch thata+b+c=1. Then
∈
theinequality
1 y
af(x)+bf(a x+(1 a )y)+cf(y) f(t)dt (20)
− ≥ y xZx
−
issatisfiedforallx,y Randallcontinuousandconvexfunctions f :[x,y] Rif,
∈ →
andonlyif,
1
b(1 a )+c= (21)
− 2
andoneofthefollowingconditionsholdstrue:
(i)a+a 1,
(ii)a+b≥+a 1,
(iii)a+a <1≤,a+b+a >1and2a+a 1.
≥
Note that the original Hermite-Hadamard inequality consists of two parts. We
treatedthesecasesseparately.However,itispossibletoformulatearesultcontaining
bothinequalities.
Corollary1 ([40]).Ifa,a ,b (0,1)satisfy(19)andoneoftheconditions(i),(ii),
∈
(iii)ofTheorem4,thentheinequality
1 y
af(a x+(1 a y)+(1 a)f(b x+(1 b )y) f(t)dt
− − − ≤ y xZx ≤
−
(1 a )f(x)+(a b )f(ax+(1 a)y)+b f(y)
− − −
issatisfiedforallx,y Randforallcontinuousandconvexfunctions f :R R.
∈ →
Aswecansee,theOhlinlemmaisveryuseful,however,itisworthnoticingthat
in the case of some inequalities, the distribution functions cross more than once.
ThereforeasimpleapplicationoftheOhlinlemmaisimpossible.
Inthepapers[32,41],theauthorsusedtheLevin-Stecˇkintheorem[25](seealso
[30], Theorem 4.2.7), which gives necessary and sufficient conditions for convex
ordering of functions with bounded variation, which are distribution functions of
signedmeasures.
Theorem6 (Levin,Stecˇkin[25]).Leta,b R,a<bandletF ,F : [a,b] Rbe
1 2
∈ →
functionswithboundedvariationsuchthatF (a)=F (a).Then,inorderthat
1 2
b b
f(x)dF (x) f(x)dF (x) (22)
1 2
Za ≤Za
10 TeresaRajba
forallcontinuousconvexfunctions f: [a,b] R,itisnecessaryandsufficientthat
→
F andF verifythefollowingthreeconditions:
1 2
F (b)= F (b), (23)
1 2
b b
F (x)dx = F (x)dx, (24)
1 2
Za Za
x x
F (t)dt F (t)dt forall x (a,b). (25)
1 2
Za ≤ Za ∈
Definethenumberofsignchangesofafunctionj : R Rby
→
S (j )=sup S [j (x ),j (x ),...,j (x )]: x <x <...x R,k N ,
− − 1 2 k 1 2 k
{ ∈ ∈ }
where S [y ,y ,...,y ] denotes the number of sign changes in the sequence y ,
− 1 2 k 1
y ,..., y (zero terms are being discarded). Two real functions j ,j are said
2 k 1 2
to have n crossing points (or cross each other n-times) if S (j j )= n. Let
− 1 2
a=x <x <...<x <x =b.Wesaythatthefunctionsj ,j −crossesn-times
0 1 n n+1 1 2
at the pointsx ,x ,...,,x (or that x ,x ,...,,x are the points of sign changesof
1 2 n 1 2 n
j j )ifS (j j )=nandthereexista<x <x <...<x <x <x <b
1 2 − 1 2 1 1 n n n+1
suc−hthatS [x ,x −,...,x ]=n.
− 1 2 n+1
Szostok[41]usedTheorem6tomakeanobservation,whichismoregeneralthan
Ohlin’slemmaandconcernsthesituationwhenthefunctionsF andF havemore
1 2
crossing points than one. In [41] is given some useful modification of the Levin-
Stecˇkintheorem[25],whichcanberewritteninthefollowingform.
Lemma2([41]). Let a,b R, a<b and let F ,F : (a,b) R be functions with
1 2
∈ →
boundedvariationsuchthatF(a)=F(b)=0, bF(x)dx=0,whereF =F F .
a 2− 1
Leta<x1<...<xm<bbethepointsofsignRchangesofthefunctionF.Assume
thatF(t) 0fort (a,x ).
1
≥ ∈
Ifmiseventhentheinequality
•
b b
f(x)dF (x) f(x)dF (x) (26)
1 2
Za ≤Za
isnotsatisfiedbyallcontinuousconvexfunctions f: [a,b] R.
→
Ifmisodd,defineA (i=0,1,...,m,x =a,x =b)
i 0 m+1
•
xi+1
A = F(x)dx.
i
Zxi | |
Thentheinequality(26)issatisfiedforallcontinuousconvexfunctions f: [a,b]
R,if,andonlyif,thefollowinginequalitiesholdtrue: →