GENERALIZATION OF MAJORIZATION THEOREM VIA ABEL-GONTSCHAROFF POLYNOMIAL ...

26 Pages·2015·0.25 MB·English

RADHAZU.MATEMATIČKEZNANOSTI
Vol. 19=523(2015): 91-116
GENERALIZATION OF MAJORIZATION THEOREM VIA
ABEL-GONTSCHAROFF POLYNOMIAL
Muhammad Adil Khan, Naveed Latif and Josip Pečarić
Abstract. In this paper we use Abel-Gontscharoff formula and
Green function to give some identities for the difference of majorization
inequality and present the generalization of majorization theorem for the
classofn-convex. WeuseinequalitiesfortheČebyševfunctionaltoobtain
boundsfortheidentitiesrelatedtogeneralizationsofmajorizationinequal-
ities. Wepresentmeanvaluetheoremsandn-exponentialconvexityforthe
functionalobtainedfromthegeneralizedmajorizationinequalities. Atthe
end we discuss the results for particular families of functions and give
means.
1. Introduction
For fixed m 2 let
≥
x=(x ,...,x ), y=(y ,...,y )
1 m 1 m
denote two real m-tuples. Let
x x ... x , y y ... y ,
[1] [2] [m] [1] [2] [m]
≥ ≥ ≥ ≥ ≥ ≥
x x ... x , y y ... y
(1) (2) (m) (1) (2) (m)
≤ ≤ ≤ ≤ ≤ ≤
be their ordered components.
Definition 1.1. [23, p. 319] x is said to majorize y (or y is said to be
majorized by x), in symbol, x y, if
≻
l l
(1.1) y x
[i] [i]
≤
i=1 i=1
X X
2010 Mathematics Subject Classification. Primary26D15,Secondary 26D20.
Key words and phrases. Majorizationinequality,Abel-Gontscharoffformula,Čebyšev
functional,Ostrowski-typeinequality,n-exponential convexity.
Theresearchofthethirdauthor hasbeensupportedbyCroatianScience Foundation
undertheproject5435.
91
92 M. ADIL KHAN, N. LATIF AND J. PEČARIĆ
holds for l=1,2,...,m 1 and
−
m m
x = y .
i i
i=1 i=1
X X
Note that (1.1) is equivalent to
m m
y x
(i) (i)
≤
i=m l+1 i=m l+1
X− X−
holds for l=1,2,...,m 1.
−
The following theorem is well-known as the majorization theorem given by
Marshall and Olkin [20, p. 14] (see also [23, p. 320]):
Theorem 1.2. Let x=(x ,...,x ),y=(y ,...,y ) be two m-tuples such
1 m 1 m
that x , y [a,b] (i=1,...,m). Then
i i
∈
m m
(1.2) φ(y ) φ(x )
i i
≤
i=1 i=1
X X
holds for every continuous convex function φ : [a,b] R if and only if x
→ ≻
y holds.
The followingtheoremcanbe regardedas aweightedversionofTheorem
1.2 and is proved by Fuchs in [14] ([20, p. 580], [23, p. 323]):
Theorem 1.3. Let x = (x ,...,x ),y = (y ,...,y ) be two decreasing
1 m 1 m
real m-tuples with x , y [a,b] (i = 1,...,m) and w = (w ,w ,...,w ) be a
i i 1 2 m
∈
real m-tuple such that
l l
(1.3) w y w x for l=1,...,m 1,
i i i i
≤ −
i=1 i=1
X X
and
m m
(1.4) w y = w x .
i i i i
i=1 i=1
X X
Then for every continuous convex function φ:[a,b] R, we have
→
m m
(1.5) w φ(y ) w φ(x ).
i i i i
≤
i=1 i=1
X X
ThefollowingintegralversionofTheorem1.3isasimpleconsequenceofTheo-
rem 12.14 in [22] (see also [23, p.328]):
Theorem 1.4. Let x,y : [a,b] [α,β] be decreasing and w : [a,b] R
→ →
be continuous functions. If
ν ν
(1.6) w(t)y(t)dt w(t)x(t)dt for every ν [a,b],
≤ ∈
Za Za
GENERALIZATION OF MAJORIZATION THEOREM 93
and
b b
(1.7) w(t)y(t)dt = w(t)x(t)dt
Za Za
hold, then for every continuous convex function φ:[α,β] R, we have
→
b b
(1.8) w(t)φ(y(t)) dt w(t)φ(x(t)) dt.
≤
Za Za
Forsomeotherrelatedresultsandgeneralizationofmajorizationtheorem
see [20, p. 583], [1]-[6], [8,13,18,19,21].
Consider the Green function G defined on [α,β] [α,β] by
×
(t−β)(s−α) , α s t;
(1.9) G(t,s)= β α ≤ ≤
((s−β−)(t−α), t s β.
β α ≤ ≤
−
The function G is convex in s, it is symmetric, so it is also convex in t. The
function G is continuous in s and continuous in t.
For any function φ : [α,β] R, φ C2([α,β]), we can easily show by
→ ∈
integrating by parts that the following is valid
β x x α β
(1.10) φ(x)= − φ(α)+ − φ(β)+ G(x,s)φ′′(s)ds,
β α β α
− − Zα
where the function G is defined as above in (1.9) ([26]).
In this paper, n always denotes a positive integer number. Throughout,
in what follows, we shall assume that the function φ that is n-times contin-
uously differentiable on the interval [α,β] (i.e., φ Cn[α,β]), although this
∈
restriction is not necessary.
The Abel-Gontscharoff interpolation problem in the real case was intro-
duced in 1935 by Whittaker [26] and subsequently by Gontscharoff [15] and
Davis[12]. ThefollowingtheoremisAbel-Gontscharofftheoremfortwopoints
with integral remainder.
Theorem 1.5 ([7]). Let n,k N, n 2, 0 k n 1, and φ
∈ ≥ ≤ ≤ − ∈
Cn([α,β]). Then we have
(1.11) φ(s)=ρ (α,β,φ,s)+R(φ,s)
n 1
−
where ρ (α,β,φ,s) is the Abel-Gontscharoff interpolating polynomial for
n 1
−
two points of degree n 1, i.e.
−
k (s α)i
ρ (α,β,φ,s)= − φ(i)(α)
n 1
− i!
i=0
X
+n−k−2 j (s−α)k+1+i(α−β)j−i φ(k+1+j)(β)
(k+1+i)!(j i)!
" #
j=0 i=0 −
X X
94 M. ADIL KHAN, N. LATIF AND J. PEČARIĆ
and the remainder is given by
β
R(φ,s)= G (s,t)φ(n)(t)dt
n
Zα
and G (s,t) is defined by
n
(1.12)
G (s,t)= 1 ki=0 n−i1 (s−α)i(α−t)n−i−1 , α≤t≤s;
n (n−1)!(P− ni=−(cid:0)k1+1(cid:1)n−i1 (s−α)i(α−t)n−i−1, s≤t≤β.
Further, for α s,t Pβ the fo(cid:0)llow(cid:1)ing inequalities hold
≤ ≤
∂iG (s,t)
(1.13) ( 1)n k 1 n 0, 0 i k,
− −
− ∂si ≥ ≤ ≤
∂iG (s,t)
(1.14) ( 1)n−i n 0, k+1 i n 1.
− ∂si ≥ ≤ ≤ −
In order to recall the definition of n convex function, first we write the defi-
−
nition of divided difference.
Definition 1.6. [23, p. 15] Let φ be a real-valued function defined on
[α,β]. The divided difference of order n of the function φ at distinct points
[α,β] is defined recursively by
φ[x ]=φ(x ), (i=0,...,n)
i i
and
φ[x ,...,x ] φ[x ,...,x ]
1 n 0 n 1
φ[x0,...,xn]= − − .
x x
n 0
−
The value φ[x ,...,x ] is independent of the order of the points x ,...,x .
0 n 0 n
The definitionmaybe extendedto include the casethatsome(orall)the
points coincide. Assuming that φ(j 1)(x) exists, we define
−
φ(j 1)(x)
−
(1.15) φ[x,...,x]= .
(j 1)!
−
j times
−
Definition 1.7. [23,|p.{1z5]}A function φ : [α,β] R is said to be n-
→
convex, n 0, on [α,β] if and only if for all choices of (n+1) distinct points
≥
x ,...,x [α,β], the nth order divided difference is non negative that is
0 n
∈
φ[x ,x ,...,x ] 0.
0 1 n
≥
Inthis paperweutilize Abel-Gontscharoff’stheoremwiththe integralre-
mainder and Green function to establish generalizationof majorization theo-
rem for the class of n-convex functions. We use inequalities for the Čebyšev
functionaltoobtainboundsfortheidentitiesrelatedtogeneralizationsofma-
jorization inequalities. We present mean value theorems and n-exponential
GENERALIZATION OF MAJORIZATION THEOREM 95
convexity for the functional obtained from the generalized majorization in-
equalities which leads to exponential convexity and log-convexity for these
functionals. Finally, we discuss the results for particular families of function
and give classes of Cauchy type means and prove their monotonicity.
2. Main results
We begin this section with the proof of some identities related to gener-
alizations of majorization inequality.
Theorem 2.1. Let n,k N, n 4, 0 k n 1, φ Cn([α,β]) and
∈ ≥ ≤ ≤ − ∈
w = (w ,...,w ), x = (x ,...,x ) and y = (y ,...,y ) be m-tuples such that
1 m 1 m 1 m
x , y [α,β],w R (l = 1,...,m). Also let G and G be defined by (1.9)
l l l n
∈ ∈
and (1.12) respectively. Then
m m m
φ(β) φ(α)
w φ(x ) w φ(y )= − w (x y )
l l l l l l l
− β α −
l=1 l=1 − l=1
X X X
k φ(i+2)(α) β m
+ w (G(x ,s) G(y ,s)) (s α)ids
l l l
i=0 i! Zα "l=1 − # −
X X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
(2.1) + − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β m
w (G(x ,s) G(y ,s)) (s α)k+1+ids
l l l
×Zα "l=1 − # −
X
β β m
+ w (G(x ,s) G(y ,s)) G (s,t)φ(n)(t)dtds.
l l l n 2
Zα Zα "l=1 − # −
X
Proof. Using (1.10) in m w φ(x ) m w φ(y ) we have
l=1 l l − l=1 l l
P P
m m
(2.2) w φ(x ) w φ(y )=
l l l l
−
l=1 l=1
X X
m
φ(β) φ(α)
− w (x y )
l l l
β α −
− l=1
X
β m m
+ w G(x ,s) w G(y ,s) φ (s)ds.
l l l l ′′
Zα "l=1 − l=1 #
X X
96 M. ADIL KHAN, N. LATIF AND J. PEČARIĆ
By Theorem 1.5, φ (s) can be expressed as
′′
k (s α)i
(2.3) φ′′(s)= − φ(i+2)(α)
i!
i=0
X
+n−k−4 j (s−α)k+1+i(α−β)j−i φ(k+3+j)(β)
(k+1+i)!(j i)!
" #
j=0 i=0 −
X X
β
+ G (s,t)φ(n)(t)dt.
n 2
Zα −
Using (2.3) in (2.2) we get (2.1).
Integral version of the above theorem can be stated as:
Theorem 2.2. Let n,k N, n 4, 0 k n 1, φ Cn([α,β]), and
let x,y : [a,b] [α,β], w : [a∈,b] R≥ be co≤ntinu≤ous−functio∈ns and G, G be
n
→ →
defined by (1.9) and (1.12) respectively. Then
b b
(2.4) w(τ)φ(x(τ))dτ w(τ)φ(y(τ))dτ =
−
Za Za
φ(β) φ(α) b
− w(τ)(x(τ) y(τ))dτ
β α −
− Za
k φ(i+2)(α) β b
+ w(τ)(G(x(τ),s) G(y(τ),s))dτ (s α)ids
i=0 i! Zα Za − ! −
X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
+ − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β b
w(τ)(G(x(τ),s) G(y(τ),s))dτ (s α)k+1+ids
×Zα Za − ! −
β β b
+ w(τ)(G(x(τ),s) G(y(τ),s))dτ G (s,t)φ(n)(t)dtds.
n 2
Zα Zα Za − ! −
In the following theorem we obtain generalizations of majorization in-
equality for n- convex functions.
Theorem 2.3. Let n,k N, n 4, 0 k n 1, w=(w ,...,w ), x=
1 m
(x ,...,x ) and y = (y ,...,y∈ ) be ≥m-tuple≤s su≤ch t−hat x , y [α,β],w R
1 m 1 m l l l
∈ ∈
(l = 1,...,m). Also let G and G be defined by (1.9) and (1.12) respectively.
n
If φ:[α,β] R is n convex, and
→ −
β m
(2.5) w (G(x ,s) G(y ,s)) G (s,t)ds 0, t [α,β].
l l l n 2
Zα l=1 − ! − ≥ ∈
X
GENERALIZATION OF MAJORIZATION THEOREM 97
Then
m m m
φ(β) φ(α)
w φ(x ) w φ(y ) − w (x y )
l l l l l l l
− ≥ β α −
l=1 l=1 − l=1
X X X
k φ(i+2)(α) β m
+ w (G(x ,s) G(y ,s)) (s α)ids
l l l
i=0 i! Zα "l=1 − # −
(2.6) X X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
+ − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β m
w (G(x ,s) G(y ,s)) (s α)k+1+ids.
l l l
×Zα"l=1 − # −
X
Ifthereverseinequalityin(2.5) holds, thenalsothereverseinequalityin(2.6)
holds.
Proof. Since the function φ is n convex, therefore without loss of ge-
−
nerality we can assume that φ is n times differentiable and φ(n)(x) 0 for
− ≥
all x [α,β] (see [23, p. 16 and p. 293]). Hence, we can apply Theorem 2.1
∈
to obtain (2.6).
Remark 2.1. Asfrom(1.13)wehave( 1)n k 3G (s,t) 0,therefore
− − n 2
− − ≥
forthecasewhennisevenandk isoddornisoddandk iseven,itisenough
to assume that m w G(x ,s) m w G(y ,s) 0,s [α,β], instead of
l=1 l l − l=1 l l ≥ ∈
the assumption(2.5)inTheorem2.3. Similarlywecandiscussforthe reverse
P P
inequality in (2.6).
Integral version of the above theorem can be stated as:
Theorem 2.4. Let n,k N, n 4, 0 k n 1, x,y : [a,b] [α,β],
w : [a,b] R be continuou∈s functi≥ons an≤d G,≤G −be defined by (→1.9) and
n
(1.12) resp→ectively. If φ:[α,β] R is n convex, and
→ −
β b
(2.7) w(τ)(G(x(τ),s) G(y(τ),s))dτ G (s,t)ds 0.
n 2
Zα Za − ! − ≥
98 M. ADIL KHAN, N. LATIF AND J. PEČARIĆ
Then
b b
(2.8) w(τ)φ(x(τ))dτ w(τ)φ(y(τ))dτ
− ≥
Za Za
φ(β) φ(α) b
− w(τ)(x(τ) y(τ))dτ
β α −
− Za
k φ(i+2)(α) β b
+ w(τ)(G(x(τ),s) G(y(τ),s))dτ (s α)ids
i=0 i! Zα Za − ! −
X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
+ − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β b
w(τ)(G(x(τ),s) G(y(τ),s))dτ (s α)k+1+ids.
×Zα Za − ! −
Ifthereverseinequalityin(2.7) holds, thenalsothereverseinequalityin(2.8)
holds.
Remark 2.2. Asfrom(1.13)wehave( 1)n k 3G (s,t) 0,therefore
− − n 2
− − ≥
forthecasewhennisevenandk isoddornisoddandk iseven,itisenough
b
to assume that w(τ)(G(x(τ),s) G(y(τ),s))dτ 0,s [α,β], instead of
a − ≥ ∈
the assumption(2.7)inTheorem2.4. Similarlywecandiscussforthe reverse
R
inequality in (2.8).
We give generalization of majorization theorem for majorized m-tuples:
Theorem 2.5. Let n,k N, n 4, 0 k n 1 and x= (x ,...,x ),
1 m
∈ ≥ ≤ ≤ −
y = (y ,...,y ) be two m-tuples such that y x with x , y [α,β], (l =
1 m l l
1,...,m). AlsoletGbedefinedby(1.9). Consid≺er φ:[α,β] R∈is n convex.
→ −
(i) If n is even and k is odd or n is odd and k is even. Then
m m
(2.9) φ(x ) φ(y )
l l
− ≥
l=1 l=1
X X
k φ(i+2)(α) β m
w (G(x ,s) G(y ,s)) (s α)ids
l l l
i=0 i! Zα "l=1 − # −
X X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
+ − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β m
w (G(x ,s) G(y ,s)) (s α)k+1+ids.
l l l
×Zα "l=1 − # −
X
GENERALIZATION OF MAJORIZATION THEOREM 99
Moreover if φ(i+2)(α) 0 for i = 0,...,k and φ(k+3+j)(β) 0 if
≥ ≥
j i is even and φ(k+3+j)(β) 0 if j i is odd for i = 0,...,j and
− ≤ −
j = 0,...,n k 4, then the right hand side of (2.9) will be non
− −
negative, that is (1.2) holds.
(ii) Ifnandk bothareevenorbothareodd,thenreverseinequalityholdsin
(2.9). Moreover if φ(i+2)(α) 0 for i= 0,...,k and φ(k+3+j)(β) 0
≤ ≤
if j i is even and φ(k+3+j)(β) 0 if j i is odd for i=0,...,j and
− ≥ −
j =0,...,n k 4,thentheright handsideof thereverseinequalityin
− −
(2.9) will be non positive, that is the reverse inequality in (1.2) holds.
Proof. Byusing(1.13)wehave( 1)n k 3G (s,t) 0, α s,t β,
− − n 2
− − ≥ ≤ ≤
thereforeifnisevenandkisoddornisoddandkiseventhenG (s,t) 0.
n 2
− ≥
Also as G is convex so by Theorem 1.2 and non negativity of G , the
n 2
−
inequality (2.5) holds for w = 1, l = 1,2,..,m. Hence by Theorem 2.3 for
l
w =1,l=1,2,..,m,theinequality(2.9)holds. Byusingtheotherconditions
l
the non negativity of the right hand side of (2.9) is obvious.
Similarly we can prove (ii).
In the following theorem we present generalization of Fuchs’ majorization
theorem.
Theorem 2.6. Let n,k N, n 4, 0 k n 1, x = (x ,...,x ),
1 m
∈ ≥ ≤ ≤ −
y=(y ,...,y ) be decreasing and w=(w ,...,w ) be any m-tuples such that
1 m 1 m
x , y [α,β],w R (l = 1,...,m) which satisfies (1.3) and (1.4). Also let
l l l
G be d∈efined by (1∈.9). Consider φ:[α,β] R is n convex.
→ −
(i) If n is even and k is odd or n is odd and k is even. Then
m m
(2.10) w φ(x ) w φ(y )
l l l l
− ≥
l=1 l=1
X X
k φ(i+2)(α) β m
w (G(x ,s) G(y ,s)) (s α)ids
l l l
i=0 i! Zα "l=1 − # −
X X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
+ − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β m
w (G(x ,s) G(y ,s)) (s α)k+1+ids.
l l l
×Zα "l=1 − # −
X
Moreover if φ(i+2)(α) 0 for i = 0,...,k and φ(k+3+j)(β) 0 if
≥ ≥
j i is even and φ(k+3+j)(β) 0 if j i is odd for i = 0,...,j and
− ≤ −
j = 0,...,n k 4, then the right hand side of (2.10) will be non
− −
negative, that is (1.5) holds.
100 M. ADIL KHAN, N. LATIF AND J. PEČARIĆ
(ii) Ifnandk bothareevenorbothareodd,thenreverseinequalityholdsin
(2.10). Moreover if φ(i+2)(α) 0 for i=0,...,k and φ(k+3+j)(β) 0
≤ ≤
if j i is even and φ(k+3+j)(β) 0 if j i is odd for i=0,...,j and
− ≥ −
j =0,...,n k 4, then the right hand side of the reverse inequality
− −
in (2.10) will be non positive, that is the reverse inequality in (1.5)
holds.
Proof. TheproofissimilartotheproofofTheorem2.5butuseTheorem
1.3 instead of Theorem 1.2.
The integral version of Theorem 2.6 can be stated as:
Theorem 2.7. Let n,k N, n 4, 0 k n 1, x,y : [a,b] [α,β]
be decreasing and w : [a,b] ∈ R be≥any co≤ntinu≤ous−function. Also →let G be
defined by (1.9). Consider φ→:[α,β] R is n convex and
→ −
ν ν
(2.11) w(τ)y(τ)dτ w(τ)x(τ)dτ for ν [a,b],
≤ ∈
Za Za
b b
(2.12) w(τ)x(τ)dτ = w(τ)y(τ)dτ
Za Za
(i) If n is even and k is odd or n is odd and k is even. Then
(2.13)
b b
w(τ)φ(x(τ))dτ w(τ)φ(y(τ))dτ
−
Za Za
k φ(i+2)(α) β b
w(τ)(G(x(τ),s) G(y(τ),s))dτ (s α)ids
≥ i=0 i! Zα Za − ! −
X
n−k−4 j ( 1)j−i(β α)j−iφ(k+3+j)(β)
+ − −
(k+1+i)!(j i)!
j=0 i=0 −
X X
β b
w(τ)(G(x(τ),s) G(y(τ),s))dτ (s α)k+1+ids.
×Zα Za − ! −
Moreover if φ(i+2)(α) 0 for i = 0,...,k and φ(k+3+j)(β) 0 if
≥ ≥
j i is even and φ(k+3+j)(β) 0 if j i is odd for i = 0,...,j and
− ≤ −
j = 0,...,n k 4, then the right hand side of (2.13) will be non
− −
negative, that is integral version of (1.5) holds.
(ii) Ifnandk bothareevenorbothareodd,thenreverseinequalityholdsin
(2.13). Moreover if φ(i+2)(α) 0 for i=0,...,k and φ(k+3+j)(β) 0
≤ ≤
if j i is even and φ(k+3+j)(β) 0 if j i is odd for i=0,...,j and
− ≥ −
j =0,...,n k 4,thentheright handsideof thereverseinequalityin
− −