Weighted majorization inequalities for nconvex functions via extension of Montgomery identity using Green function
Weighted majorization inequalities for n convex functions via extension of Montgomery identity using Green function
Andrea Aglic´ Aljinovic´ 0 1 2
Asif R. Khan 0 1 2
Josip E. Pecˇaric´ 0 1 2
Mathematics Subject Classification 0 1 2
0 J. E. Pecˇaric ́ Faculty of Textile Technology, University of Zagreb , Prilaz Baruna Filipovic ́a 28A, 10000 Zagreb , Croatia
1 A. R. Khan Department of Mathematics, Faculty of Science, University of Karachi , University Road, Karachi 75270 , Pakistan
2 A. Aglic ́ Aljinovic ́ (
New identities and inequalities are given for weighted majorization theorem for nconvex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Cˇ ebyšev type inequalities. Mean value theorems are also discussed for functional related to new results.

26A51 · 26D15 · 26D20
1 Introduction
Convex functions have got attention of most of the mathematicians of present era (see, e.g.) [
7, 15–18, 20–
22, 24
]. For a standard definition of convex function and some alternate forms, we refer the reader to the
following books [
12, 14, 19, 23
]. Now, we commence the present article by recalling related concepts and
definition of generalized convex function as in (see, e.g.) [
2, 6, 9, 10
] and [
19
].
Definition 1.1 The nth order divided difference of a realvalued function f defined on an interval I of R at
distinct points ξi , ξi+1, . . . , ξi+n for i ∈ N is defined recursively by
ξ j ; f
= f ξ j , j ∈ {i, . . . , i + n}
ξi , . . . , ξi+n ; f =
ξi+1, . . . , ξi+n ; f
− ξi , . . . , ξi+n−1; f
ξi+n − ξi
.
Remark 1.2 Let us denote [ξi , . . . , ξi+n ; f ] by (n) f (ξi ). The value ξi , . . . , ξi+n ; f is independent of the
order of the points ξi , ξi+1, . . . , ξi+n . We can extend this definition by considering the cases in which two or
more points coincide by taking respective limits.
Definition 1.3 A function f : I →
distinct points ξi , . . . , ξi+n we have
R is called convex of order n or nconvex if for all choices of (n + 1)
(n) f (ξi ) ≥ 0.
· · · ≤ ξ[
2
] ≤ ξ[
1
],
If nth order derivative f (n) exists, then f is nconvex iff f (n) ≥ 0.
For fixed m ≥ 2, let x = (ξ1, . . . , ξm ) and y = (η1, . . . , ηm ) denote two real mtuples and ξ[m] ≤ ξ[m−1] ≤
η[m] ≤ η[m−1] ≤ · · · ≤ η[
2
] ≤ η[
1
] their ordered components.
Definition 1.4 For x, y ∈ Rm ,
x ≺ y if
k
im=1 ξ[i] ≤
i=1 ξi =
k
mi=1 η[i],
i=1 ηi ,
k ∈ {1, . . . , m − 1},
when x ≺ y, x is said to be majorized by y.
This concept and notation of majorization were introduced by Hardy, Littlewood and Pólya [
5
]. Next, we
state the wellknown majorization theorem from [
5
].
Proposition 1.5 Let x, y ∈ Rm . The inequality
m
i=1
f (ξi ) ≤
f (ηi )
m
i=1
holds for any continuous convex function f : R → R if and only if x ≺ y.
The following weighted version of the majorization theorem was given by Fuchs in [
4
] (see also [11, p.
580] and [19, p. 323]).
Proposition 1.6 Let p ∈ Rm and let x, y be two nonincreasing real mtuples such that
(1.1)
(1.2)
(1.3)
(1.4)
Then, for any continuous convex function f : R → R, the following inequality holds
and
k
i=1
m
i=1
pi ξi ≤
pi ξi =
m
i=1
k
i=1
m
i=1
pi ηi .
m
i=1
pi f (ξi ) ≤
pi f (ηi ).
pi ηi , k ∈ {1, . . . , m − 1}
Some integral inequalities in relation to majorization can be stated as follows. The following proposition
is a consequence of Theorem 1 in [
16
] (see also [19, p. 328]).
Proposition 1.7 Let ξ, η : [α0, β0] → R be two nonincreasing continuous functions and let p : [α0, β0] →
be a continuous function. Then, if
p (u) ξ(u) du ≤
p (u) η(u) du, for each γ ∈ (α0, β0),
and
p (u) ξ(u) du ≤
p (u) ξ(u) du =
p (u) η(u) du, for each γ ∈ (α0, β0),
p (u) η(u) du,
then again inequality (1.7) holds.
In this article, we will state our results for nonincreasing x and y satisfying the assumption of Proposition
1.7, but they are still valid for nondecreasing x and y satisfying the above condition see for example [11,
p. 584].
From [
1
], we have extracted here an extension of Montgomery identity via Taylor’s formula which may
be stated as follows.
Proposition 1.9 Let n ∈ N, f : I → R be such that f (n−1) is absolutely continuous, I ⊂ R an open interval,
a0, b0 ∈ I , a0 < b0. Then, the following identity holds
and
p (u) ξ(u) du =
p (u) η(u) du,
hold, then for every continuous convex function f : R → R the following inequality holds
β0
α0
p (u) f (ξ(u)) du ≤
p (u) f (η(u)) du.
β0
α0
Remark 1.8 Let ξ, η : [α0, β0] → R be two nondecreasing continuous functions and let p : [α0, β0] → be a
continuous function. If
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
where
In case n = 1, the sum
(see for instance [
13
]),
f (x) = b0 − a0 a0
1
b0
f (s) ds +
n−2 f (k+1) (b0) (x − b0)k+2
−
k=0 k! (k + 2)
n−2 f (k+1) (a0) (x − a0)k+2
k=0 k! (k + 2)
⎪⎧ − n((xb−0−s)an0) + bx0−−aa00 (x − s)n−1 , a0 ≤ s ≤ x,
Tn (x, s) = ⎨
(x−s)n
⎪⎩ − n(b0−a0) + bx0−−ba00 (x − s)n−1 , x < s ≤ b0.
n−2
k=0 · · · is empty, so identity (1.8) reduces to the famous Montgomery identity
where P (x, s) is the Peano kernel, defined by
f (s) = b0 − a0 a0
1
b0
f (t) dt +
b0
a0
P (s, t) f (t) dt
⎧ bt0−−aa00 , a0 ≤ t ≤ s,
P (s, t) = ⎨
⎩ bt0−−ba00 , s < t ≤ b0.
Finally, we define of Green function G : [a0, b0] × [a0, b0] by
G(s, t ) =
(s−b0)(t−a0)
b0−a0
(t−b0)(s−a0)
b0−a0
for a0 ≤ t ≤ s,
for s ≤ t ≤ b0.
The function G is continuous and convex with respect to both t and s.
For any function f : [a0, b0] → R, f ∈ C 2[a0, b0], we can obtain the following integral identity by
simply using integration by parts
b0 − x x − a0 f (b0) +
f (x ) = b0 − a0 f (a0) + b0 − a0
b0
a0
G(x , s) f (s)ds,
(1.11)
where the function G is defined as above in (1.10) (see also [
24
]).
The aim of the present article is to introduce new weighted majorization theorems for convex functions
via extension of Montgomery identity.
2 Majorization type identities and inequalities via extension of Montgomery identity
In this section, we state results related to weighted majorization identities and inequalities. First, we define
notations in terms of positive linear functional:
( pi , ξi , ηi , f ) =
f (b0) − f (a0)
where pi , ξi , ηi and f are defined in Proposition 1.6 and G is defined in (1.10). Also
Theorem 2.1 Let all the assumptions of Proposition 1.6 be valid. Also let f : I → R be a function such that
f (n−1) (n ≥ 3) is absolutely continuous, I ⊂ R an open interval, a0, b0 ∈ I , a0 < b0. Then for all s ∈ [a0, b0]
we have the following identity:
(1.10)
(2.1)
(2.2)
(2.3)
(2.4)
ds
(2.5)
where
⎧ 1
T˜n−2 (s, t ) = ⎨⎪⎪ b0−a0
and G(·, s) is as defined in (1.10). We also have the following identity:
where Tn is defined in Proposition 1.9.
Proof Using (1.11) in (2.1) and using linearity of ( pi , ξi , ηi , f ), we get
Differentiating (1.8) twice with respect to s, we get
b0
a0
f (s) =
b0
b0
Similarly, using (2.9) in (2.7) and applying Fubini’s theorem, we get (2.6).
( pi , ξi , ηi , G(·, s))Tn−2(s, t )ds dt
f k (a0)(s − a0)k−1 − f k (b0)(s − b0)k−1
T˜n−2(s, t ) f (n)(t )dt ds
( pi , ξi , ηi , f ) ≥
( pi , ξi , ηi , f ) ≥
+
+
+
+
b0
a0
b0
a0
( pi , ξi , ηi , id)
b0
b0
a0
Theorem 2.2 Let all the assumptions of Theorem 2.1 be valid with additional condition
where G is defined in (1.10) and T˜n in Theorem 2.1. Then for every nconvex function f : I → R, the following
inequality holds:
( pi , ξi , ηi , G(·, s))T˜n−2(s, t )ds ≥ 0, ∀ t ∈ [a0, b0]
(2.10)
ds.
(2.11)
(2.12)
ds.
(2.13)
(2.14)
Proof Function f is nconvex, so we have f (n) ≥ 0. Using this fact and (2.10) in (2.5) we obtain required
result.
Theorem 2.3 Let all the assumptions of Theorem 2.1 hold with additional condition
( pi , ξi , ηi , G(·, s))Tn−2(s, t )ds ≥ 0, ∀ t ∈ [a0, b0]
where G is as defined in (1.10) and Tn as in Proposition 1.9. Then for every nconvex function f : I → R,
the following inequality holds
f k (a0)(s − a0)k−1 − f k (b0)(s − b0)k−1
Proof Function f is nconvex, so we have f (n) ≥ 0. Using this fact and (2.12) in (2.6), we obtain required
result.
Next, we state an important consequence.
Theorem 2.4 Let all the assumptions from Theorem 2.1 hold. If f is nconvex and n is even, then inequalities
(2.11) and (2.13) hold.
Proof Since Green function G(s, t ) is convex with respect to t for all s ∈ [a0, b0] and x = (ξ1, . . . , ξm ) and
p = ( p1, . . . , pm ) satisfy condition (1.2) and (1.3). Therefore, from Proposition 1.6 (inequality (1.4)), we have
( pi , ξi , ηi , G(·, s)) ≥ 0 for s ∈ [a0, b0].
Also note that T˜n−2(s, t ) ≥ 0 (Tn−2(s, t ) ≥ 0) if n − 2 is even. Therefore combining this fact with (2.14),
we get inequality (2.10) (inequality (2.12)). As f is nconvex, results follow from Theorem 2.2 (Theorem 2.3,
respectively).
Following is the integral version of our main results. Since proofs of these results are of similar nature we
omit details.
have the following identity
Theorem 2.5 Let all the assumptions of Proposition 1.7 be valid. Let f : I → R be a function such that
f (n−1) is absolutely continuous, I ⊂ R an open interval, a0, b0 ∈ I , a0 < b0. Then for all s ∈ [a0, b0] we
where G is as defined in (1.10) and T˜n is defined in Theorem 2.1. Then, for every nconvex function f : I → R,
the following inequality holds
Theorem 2.6 Let all the assumptions of Theorem 2.5 hold with additional condition
( p, ξ, η, f ) ≥
b0 − a0
+
+
n−1
k=2
Theorem 2.7 Let all the assumptions of Theorem 2.5 hold with additional condition
( p, ξ, η, G(·, s))Tn−2(s, t)ds ≥ 0, ∀ t ∈ [a0, b0]
3 Bounds for identities related to generalized linear inequalities
Let g, h : [a0, b0] → R be two Lebesgue integrable functions. We consider the Cˇebyšev functional defined by
1
T (g, h) = b0 − a0 a0
b0
g(u)h(u)du −
1
Proposition 3.1 Let g : [a0, b0] → R be a Lebesgue integrable function and let h : [a0, b0] → R be an
absolutely continuous function with (· − a0)(b0 − ·)[h ]2 ∈ L[a0, b0]. Then we have the inequality
1 1
T (g, h) ≤ √2 b0 − a0 T (g, g)
b0
a0
(u − a)(b − u)[h (u)]2du
1/2
.
The constant √1 in (3.2) is the best possible.
2
Using aforementioned result, we are going to obtain generalizations of the result proved in previous section.
Under the assumptions of Theorems 2.2, 2.3, 2.6 and 2.7, respectively, we define the following linear
functionals
1(t) =
2(t) =
3(t) =
4(t) =
b0
a0
b0
a0
b0
a0
b0
a0
( pi , ξi , ηi , G(·, s))T˜n−2(s, t)ds ≥ 0, ∀ t ∈ [a0, b0]
( pi , ξi , ηi , G(·, s))Tn−2(s, t)ds ≥ 0, ∀ t ∈ [a0, b0]
( p, ξ, η, G(·, s))T˜n−2(s, t)ds, ∀ t ∈ [a0, b0]
( p, ξ, η, G(·, s))Tn−2(s, t)ds, ∀ t ∈ [a0, b0]
Hence using these notations, we may define Cˇebyšev functional as follows (e.g., using ):
1
T ( , ) = b0 − a0 a0
b0
2(s) ds −
1
b0
ds
(2.20)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
Theorem 3.2 Let f : [a0, b0] → R be such that f (n) is an absolutely continuous function for n ∈ N with
(· − a0)(b0 − ·)[ f (n+1)]2 ∈ L[a0, b0]. Let pi and ξi , i ∈ {1, . . . , m} satisfy the assumptions of Proposition 1.6
and the functions G, T and 1 be defined in (1.10), (3.1) and (3.3), respectively. Then, we have
f k (a0)(s − a0)k−1 − f k (b0)(s − b0)k−1
1(s)ds + Rn1( f ; a0, b0),
b0
a0
b0
where the remainder Rn1( f ; a0, b0) satisfies the estimation
1
Rn1( f ; a0, b0) ≤ (n − 3)!
Therefore, we have
1
b0
(n − 3)! a0
1(t ) f (n)(t )dt =
f (n−1)(b0) − f (n−1)(a0)
where Rn1( f ; a0, b0) satisfies inequality (3.8). Now from identity (2.5), we obtain (3.7).
Theorem 3.3 Let f : [a0, b0] → R be such that f (n) is an absolutely continuous function for n ∈ N with
(· − a0)(b0 − ·)[ f (n+1)]2 ∈ L[a0, b0]. Let pi and ξi , i ∈ {1, . . . , m} satisfy the assumptions of Proposition 1.6
and the functions G, T and 2 be defined in (1.11), (3.1) and (3.4), respectively. Then, we have
ds
(3.7)
(3.8)
ds
(3.9)
1/2
.
(3.10)
( pi , ξi , ηi , f ) =
where the remainder Rn2( f ; a0, b0) satisfies the estimation
1
Rn2( f ; a0, b0) ≤ (n − 3)!
Finally, we obtain Ostrowski type inequalities for functions f whose nth derivative belongs to L p spaces.
The symbol L p [a0, b0] (1 ≤ p < ∞) denotes the space of ppower integrable functions on the interval
[a0, b0] equipped with the norm
and L∞ [a0, b0] denotes the space of essentially bounded functions on [a0, b0] with the norm
φ p =
b0
a0
φ (t ) p dt
ds
ds
(3.11)
(3.12)
ab00 λ(t )r dt
1/r
, we find a function f for which the equality in
f (n)(t ) = sgnλ(t ) · λ(t )1/(q−1).
f (n)(t ) = sgnλ(t ).
Now, we state some Ostrowski type inequalities related to the generalized majorization inequalities.
Theorem 3.4 Let all the assumptions of Theorem 2.1 be valid. Furthermore, let (q, r ) be a pair of conjugate
exponents and f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have
The constant on the R.H.S. of (3.11) is sharp for 1 < q ≤ ∞ and the best possible for q = 1.
Proof Let us denote 1
Using the identity (2.5) and applying Hölder’s inequality, we obtain
For q = ∞, take f such that
The fact that (3.11) is the best possible for q = 1 can be proved as in [8, Thm 12].
Theorem 3.5 Let all the assumptions of Theorem 2.1 be valid. Furthermore, let (q, r ) be a pair of conjugate
exponents. Let f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have
( pi , ξi , ηi , G(·, s))Tn−2(s, t )ds
The constant on the R.H.S. of (3.13) is sharp for 1 < q ≤ ∞ and the best possible for q = 1.
Finally, we state all the integral analogous of results from this section. Since proofs are of similar nature,
so we omit the details.
Theorem 3.6 Let f : [a0, b0] → R be such that f (n) is an absolutely continuous function for n ∈ N with
(· − a0)(b0 − ·)[ f (n+1)]2 ∈ L[a0, b0] and let p, x and y be as defined in Proposition 1.7 and let the functions
G, T and 3 be defined in (1.10), (3.1) and (3.5), respectively. Then, we have
where the remainder Rn3( f ; a0, b0) satisfies the estimation
+
+
+
+
+
+
( p, ξ, η, G(·, s))ds
( p, ξ, η, G(·, s))ds
Theorem 3.7 Let all the assumptions of Theorem 3.6 valid. Then, we have
where
4 is as defined in (3.6) and the remainder Rn4( f ; a0, b0) satisfies the estimation
1
Rn4( f ; a0, b0) ≤ (n − 3)!
Now, we state some Ostrowski type inequalities related to the generalized majorization inequalities.
Theorem 3.8 Let all the assumptions of Theorem 2.5 hold. Furthermore, let (q, r ) be a pair of conjugate
exponents. Let f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have
The constant on the right hand side of (3.18) is sharp for 1 < q ≤ ∞ and the best possible for q = 1.
Theorem 3.9 Let all the assumptions of Theorem 2.5 hold. Furthermore, let (q, r ) be a pair of conjugate
exponents. Let f (n) ∈ Lq [a0, b0] for some n ∈ N. Then, we have
The constant on the R.H.S. of (3.19) is sharp for 1 < q ≤ ∞ and the best possible for q = 1.
4 Mean value theorems
In the last section, we give mean value theorems for new functionals related to our obtained results. Under the
assumptions of Theorem 2.2 using (2.11), Theorem 2.3 using (2.13), Theorem 2.6 using (2.18) and Theorem
2.7 using (2.20), we define the following functionals, respectively:
1( f ) =
b0 − a0
b0
( pi , ξi , ηi , G(·, s))ds
( p, ξ, η, G(·, s))
Theorem 4.1 Let f ∈ Cn[a0, b0] and let k : Cn[a0, b0] → R for k ∈ {1, 2, 3, 4} be linear functionals as
defined in ( A1), ( A2), ( A3) and ( A4), respectively. Then, there exists ξk ∈ [a0, b0] for k ∈ {1, 2, 3, 4} such
that
k ( f ) = f (n)(ξk ) k ( f0), k ∈ {1, 2, 3, 4}
(4.5)
Proof Let us define functions and
F1(x) = M f0(x) − f (x)
F2(x) = f (x) − L f0(x)
f (n)([a0, b0]) = [L , M]
L k ( f0) ≤ k ( f ) ≤ M k ( f0).
where L and M are the minimum and maximum of the image of the nth derivative of f [a0, b0], i.e.,
Then, F1 and F2 are nconvex. Hence, k (F1) ≥ 0 and k (F2) ≥ 0 and
f (n)(ξk ).
If k ( f0) = 0, then the statement obviously holds.
If k ( f0) = 0, then kk((ff0)) ∈ [L , M] = f (n)([a0, b0]), so there exist ξk ∈ [a0, b0] such that kk((ff0)) =
Applying Theorem 4.1 for function ω = k (h) f − k ( f )h, we get the following result.
Theorem 4.2 Let f, h ∈ Cn[a0, b0] and let k : Cn[a0, b0] → R for k ∈ {1, 2, 3, 4} be linear functionals as
defined in ( A1), ( A2), ( A3) and ( A4), respectively. Then, there exists ξk ∈ [a0, b0] for k ∈ {1, 2, 3, 4} such
that
k ( f )
k (h) = h(n)(ξk )
f (n)(ξk )
assuming that both the denominators are nonzero.
Remark 4.3 If the inverse of hf ((nn)) exists, then from the above mean value theorems we can give generalized
means
ξk =
f (n) −1
h(n)
k ( f )
k (h)
.
(4.6)
Remark 4.4 Using the same method as in [
8
], we can construct new families of exponentially convex functions
and Cauchy type means (see also [
2
]). Also, using the idea described in [
8
] we can obtain the results for nconvex
functions at point.
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