International Journal of Analysis and Applications

Volume 16, Number 3 (2018), 374-399

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-374

MAJORIZATION INEQUALITIES VIA PEANO’S REPRESENTATION OF HERMITE’S

POLYNOMIAL

N. LATIF1,∗, N. SIDDIQUE2 AND J. PEČARIĆ3,4

1,∗Department of General Studies, Jubail Industrial College, Jubail Industrial City 31961, Kingdom of

Saudi Arabia

2Department of Mathematics, Govt. College University, Faisalabad 38000, Pakistan

3Faculty of Textile Technology Zagreb, University of Zagreb, Prilaz Baruna Filipovića 28A, 10000 Zagreb,

Croatia

4RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia

∗Corresponding author: naveed707@gmail.com

Abstract. The Peano’s representation of Hermite polynomial and new Green functions are used to con-

struct the identities related to the generalization of majorization type inequalities in discrete as well as

continuous case. Čebyšev functional is used to find the bounds for new generalized identities and to develop

the Grüss and Ostrowski type inequalities. Further more, we present exponential convexity together with

Cauchy means for linear functionals associated with the obtained inequalities and give some applications.

Received 2018-01-05; accepted 2018-02-20; published 2018-05-02.

2010 Mathematics Subject Classification. 26D07, 26D15, 26D20, 26D99.

Key words and phrases. classical majorization theorem; Fuchs’s thorem; Peano’s representation of Hermite’s polynomial;

Green function for ’two point right focal’ problem; Čebyšev functional; Grüss type upper bounds; Ostrowski-type bounds;

n-exponentially convex function; mean value theorems; Stolarsky type means.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

374

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-374


Int. J. Anal. Appl. 16 (3) (2018) 375

1. Introduction and Preliminaries

Newton and Lagrange gave the classical methods for constructing Hermite interpolating polynomial.

Lagrange gave the method for such function f(t) is defined at the distinct increasing points a1,a2, ...,an but

Newton gave the method for such function f(t) is defined at the distinct (but not necessarily increasing)

points a1,a2, ...,an (see [3, 7]).

We start with a brief overview of divided differences and n-convex functions and give some basic results from

the majorization theory.

An nth order divided difference of a function φ : [α,β] → R at distinct points x0,x1, ...,xn ∈ [α,β] may

be defined recursively by

[xi; φ] = φ(xi), i = 0, ...,n,

[x0, ...,xn; φ] =
[x1, ...,xn; φ] − [x0, ...,xn−1; φ]

xn −x0
.

The value [x0, ...,xn; φ] is independent of the order of the points x0, ...,xn.

A function φ is n-convex on [α,β] if

[x0,x1, ...,xn; φ] ≥ 0

holds for all choices of (n + 1) distinct points xi ∈ [α,β], i = 0, ...,n.

Remark 1.1. From this definition it follows that 1-convex function is increasing function and 2-convex

function is just convex function. If φ(n) exists, then φ is n-convex iff φ(n) ≥ 0. Also, if φ is n-convex for

n ≥ 2, then φ(k) exists and φ is (n−k)-convex for 1 ≤ k ≤ n− 2. For more informations see [13].

On the basis of various applications of the divided differences, several representations have been obtained

like error representation, Cauchy’s representation, Newton’s representation and Peano’s representation. In

this paper, we give the generalized results with the connection of Peano’s representation of Hermite’s inter-

polating polynomial and newly defined Green functions.

Majorization makes precise the vague notion that the components of a vector y are ”less spread out” or

”more nearly equal” than the components of a vector x. A complete and superb reference on the subject is

the 2011 book by Marshall et al. [12].

For fixed m ≥ 2 let

x = (x1, ...,xm) , y = (y1, ...,ym)

denote two real m-tuples. Let

x[1] ≥ x[2] ≥ ... ≥ x[m], y[1] ≥ y[2] ≥ ... ≥ y[m],

x(1) ≤ x(2) ≤ ... ≤ x(m), y(1) ≤ y(2) ≤ ... ≤ y(m)

be their ordered components.



Int. J. Anal. Appl. 16 (3) (2018) 376

Definition 1.1. [13, p. 319] x is said to majorize y (or y is said to be majorized by x), in symbol, x � y,

if

l∑
i=1

y[i] ≤
l∑
i=1

x[i] (1.1)

holds for l = 1, 2, ...,m− 1 and
m∑
i=1

xi =

m∑
i=1

yi.

Note that (1.1) is equivalent to
m∑

i=m−l+1

y(i) ≤
m∑

i=m−l+1

x(i)

holds for l = 1, 2, ...,m− 1.

The following theorem is well-known as the majorization theorem given by Marshall et al. [12, p. 14] (see

also [13, p. 320]):

Theorem 1.1. Let x = (x1, ...,xm) ,y = (y1, ...,ym) be two m-tuples such that xi, yi ∈ [α,β] (i = 1, ...,m).

Then
m∑
i=1

f (yi) ≤
m∑
i=1

f (xi) (1.2)

holds for every continuous convex function f : [α,β] → R if and only if x � y holds.

The following theorem can be regarded as a weighted version of Theorem 1.1 and is proved by Fuchs in [8]

( [12, p. 580], [13, p. 323]):

Theorem 1.2. Let x = (x1, ...,xm) ,y = (y1, ...,ym) be two decreasing real m-tuples with xi, yi ∈ [α,β]

(i = 1, ...,m) and w = (w1,w2, ...,wm) be a real m-tuple such that

l∑
i=1

wi yi ≤
l∑
i=1

wi xi for l = 1, ...,m− 1, (1.3)

and
m∑
i=1

wi yi =

m∑
i=1

wi xi. (1.4)

Then for every continuous convex function f : [α,β] → R, we have

m∑
i=1

wi f (yi) ≤
m∑
i=1

wi f (xi) . (1.5)

The following integral version of Theorem 1.2 is a simple consequence of Theorem 12.14 in [15] (see also [13,

p.328]):



Int. J. Anal. Appl. 16 (3) (2018) 377

Theorem 1.3. Let x,y : [a,b] → [α,β] be decreasing and w : [a,b] → R be continuous functions. If∫ ν
a

w(t) y(t) dt ≤
∫ ν
a

w(t) x(t) dt for every ν ∈ [a,b], (1.6)

and ∫ b
a

w(t) y(t) dt =

∫ b
a

w(t) x(t) dt (1.7)

hold, then for every continuous convex function f : [α,β] → R, we have∫ b
a

w(t) f (y(t)) dt ≤
∫ b
a

w(t) f (x(t)) dt. (1.8)

Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points. For f ∈ Cn[α,β] a

unique polynomial ρH(s) of degree (n− 1) exists satisfying any of the following conditions:

Hermite conditions:

ρ
(i)
H (aj) = f

(i)(aj); 0 ≤ i ≤ kj, 1 ≤ j ≤ r,
r∑
j=1

kj + r = n. (H)

It is of great interest to note that Hermite conditions include the following particular cases:

Type (m,n−m) conditions: (r = 2, 1 ≤ m ≤ n− 1, k1 = m− 1, k2 = n−m− 1)

ρ
(i)
(m,n)

(α) = f(i)(α), 0 ≤ i ≤ m− 1,

ρ
(i)
(m,n)

(β) = f(i)(β), 0 ≤ i ≤ n−m− 1,

Two-point Taylor conditions: (n = 2m, r = 2, k1 = k2 = m− 1)

ρ
(i)
2T (α) = f

(i)(α), ρ
(i)
2T (β) = f

(i)(β), 0 ≤ i ≤ m− 1.

We have the following result from [3].

Theorem 1.4. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and

f ∈ Cn([α,β]). Then we have

f(t) = ρH(t) + RH,n(f,t) (1.9)

where ρH(t) is the Hermite interpolating polynomial, i.e.

ρH(t) =

r∑
j=1

kj∑
i=0

Hij(t)f
(i)(aj);

the Hij are fundamental polynomials of the Hermite basis defined by

Hij(t) =
1

i!

ω(t)

(t−aj)
kj+1−i

kj−i∑
k=0

1

k!

dk

dtk

(
(t−aj)

kj+1

ω(t)

)∣∣∣∣∣
t=aj

(t−aj)
k
, (1.10)



Int. J. Anal. Appl. 16 (3) (2018) 378

ω(t) =

r∏
j=1

(t−aj)
kj+1, (1.11)

and the remainder is given by

RH,n(f,t) =

∫ β
α

GH,n(t,s)f
(n)(s)ds

where GH,n(t,s) is defined by

GH,n(t,s) =




l∑
j=1

kj∑
i=0

(aj−s)n−i−1
(n−i−1)! Hij(t); s ≤ t,

−
r∑

j=l+1

kj∑
i=0

(aj−s)n−i−1
(n−i−1)! Hij(t); s ≥ t,

(1.12)

for all al ≤ s ≤ al+1; l = 0, . . . ,r with a0 = α and ar+1 = β.

Remark 1.2. In particular cases,

for type (m,n−m) conditions, from Theorem 1.4 we have

f(t) = ρ(m,n)(t) + R(m,n)(f,t) (1.13)

where ρ(m,n)(t) is (m,n−m) interpolating polynomial, i.e

ρ(m,n)(t) =

m−1∑
i=0

τi(t)f
i(α) +

n−m−1∑
i=0

ηi(t)f
i(β),

with

τi(t) =
1

i!
(t−α)i

( t−β
α−β

)n−m m−1−i∑
k=0

(
n−m + k − 1

k

)( t−α
β −α

)k
(1.14)

and

ηi(t) =
1

i!
(t−β)i

( t−α
β −α

)m n−m−1−i∑
k=0

(
m + k − 1

k

)( t−β
α−β

)k
, (1.15)

and also the remainder R(m,n)(f,t) is given by

R(m,n)(f,t) =

∫ β
α

G(m,n)(t,s)f
(n)(s)ds

with

G(m,n)(t,s) =




m−1∑
j=0

[m−1−j∑
p=0

(
n−m+p−1

p

)(
t−α
β−α

)p]
×

(t−α)j(α−s)n−j−1
j!(n−j−1)!

(
β−t
β−α

)n−m
, α ≤ s ≤ t ≤ β,

−
n−m−1∑
i=0

[n−m−i−1∑
q=0

(
m+q−1

q

)(
β−t
β−α

)q]
×

(t−β)i(β−s)n−i−1
i!(n−i−1)!

(
t−α
β−α

)m
, α ≤ t ≤ s ≤ β.

(1.16)

For Type Two-point Taylor conditions, from Theorem 1.4 we have

f(t) = ρ2T (t) + R2T (f,t) (1.17)



Int. J. Anal. Appl. 16 (3) (2018) 379

where ρ2T (t)is the two-point Taylor interpolating polynomial i.e,

ρ2T (t) =

m−1∑
i=0

m−1−i∑
k=0

(
m + k − 1

k

)[(t−α)i
i!

( t−β
α−β

)m( t−α
β −α

)k
f(i)(α)

+
(t−β)i

i!

( t−α
β −α

)m( t−β
α−β

)k
f(i)(β)

]
(1.18)

and the remainder R2T (f,t) is given by

R2T (f,t) =

∫ β
α

G2T (t,s)f
(n)(s)ds

with

G2T (t,s) =




(−1)m
(2m−1)!p

m(t,s)
m−1∑
j=0

(
m−1+j

j

)
(t−s)m−1−jqj(t,s), s ≤ t;

(−1)m
(2m−1)!q

m(t,s)
m−1∑
j=0

(
m−1+j

j

)
(s− t)m−1−jpj(t,s), s ≥ t;

(1.19)

where p(t,s) =
(s−α)(β−t)

β−α , q(t,s) = p(s,t),∀t,s ∈ [α,β].

The following Lemma describes the positivity of Green’s function (1.12) see (Beesack [4] and [Levin [16]).

Lemma 1.1. The Green’s function GH,n(t,s) has the following properties:

(i)
GH,n(t,s)
w(t)

> 0,a1 ≤ t ≤ ar,a1 ≤ s ≤ ar;

(ii) GH,n(t,s) ≤ 1(n−1)!(β−α)|w(t)|;

(iii)
∫β
α
GH,n(t,s)ds =

w(t)
n!
.

We arrange the paper in this manner, in section 2, we use Peano’s representation of Hermite interpolating

polynomial and newly defined Green functions to establish identities for majorization inequalities. We present

generalized majorization inequalities and in particular we discuss the results for (m,n − m) interpolating

polynomial and two-point Taylor interpolating polynomial. In section 3, we give bounds for the identities

related to the generalizations of majorization inequalities by using Čebyšev functionals. We also give Grüss

type inequalities and Ostrowski-type inequalities for these functionals. In section 4, we present Lagrange and

Cauchy type mean value theorems related to the defined functionals and also give n-exponential convexity

which leads to exponential convexity and then log-convexity. At the end, in section 5, we give some related

analytical inequalities to our generalized results of upper bounds and also construct examples of exponentially

convex functions.



Int. J. Anal. Appl. 16 (3) (2018) 380

2. Main Results via Peano’s representation and new Green functions

As mentioned in [11], the complete reference about Abel-Gontscharoff polynomial and theorem for ’two-

point right focal’ problem is given in [3]:

Remark 2.1. As a special choice the Abel-Gontscharoff polynomial for ’two-point right focal’ interpolating

polynomial for n = 2 can be given as:

f(z) = f(α) + (z −α) f′(β) +
∫ β
α

GΩ,2(z,w)f
′′(w)dw, (2.1)

where GΩ,2(z,w) is the Green’s function for ’two-point right focal problem’ given as

G1(z,w) = GΩ,2(z,w) =


 (α−w) , α ≤ w ≤ z,(α−z) , z ≤ w ≤ β. (2.2)

Mehmood et al. (2017) [11] introduced some new types of Green functions by keeping in view Abel-

Gontscharoff Green’s function for ’two-point right focal problem’ that are:

G2(z,w) =


 (z −β) , α ≤ w ≤ z,(w −β) , z ≤ w ≤ β. (2.3)

G3(z,w) =


 (z −α) , α ≤ w ≤ z,(w −α) , z ≤ w ≤ β. (2.4)

G4(z,w) =


 (β −w) , α ≤ w ≤ z,(β −z) , z ≤ w ≤ β. (2.5)

Mehmood et al. (2017) gave the following lemma, using this we obtain the new generalizations of majorization

inequality.

Lemma 2.1. Let f : [α,β] → R be a twice differentiable function and Gc, (c = 1, 2, 3, 4) be the new Green

functions defined above, then along with (2.1) the following identities holds:

f(z) = f(β) + (z −β)f′(α) +
∫ β
α

G2(z,w)f
′′(w)dw, (2.6)

f(z) = f(β) − (β −α)f′(α) + (z −α)f′(α) +
∫ β
α

G3(z,w)f
′′(w)dw, (2.7)

f(z) = f(α) − (β −α)f′(α) − (β −z)f′(β) +
∫ β
α

G4(z,w)f
′′(w)dw. (2.8)

Equivalent statements between classical weighted majorization inequality and the inequality constructed

by newly Green functions are given as:



Int. J. Anal. Appl. 16 (3) (2018) 381

Theorem 2.1. Let x = (x1, ...,xm), y = (y1, ...,ym) ∈ Im be two decreasing m-tuples and also w =

(w1, ...,wm) be a real m-tuple such that satisfying (1.4) and Gc (c = 1, 2, 3, 4) is defined as in (2.2)-(2.5)

respectively. Then the following statements are equivalent:

(i) For every continuous convex function f : [α,β] → R, then
m∑
l=1

wl f (yl) ≤
m∑
l=1

wl f (xl) . (2.9)

(ii) For s ∈ [α,β], the following inequality holds
m∑
l=1

wl Gc (yl,s) ≤
m∑
l=1

wl Gc (xl,s) , c = 1, 2, 3, 4. (2.10)

Moreover, the statements (i) and (ii) are also equivalent if we change the sign of inequality in both

inequalities, in (2.9) and (2.10).

Proof. ”(i) ⇒ (ii)” Suppose the statement (i) satisfies. Fix c = 1, 2, 3, 4, the functions Gc(.,s) (s ∈ [α,β])

are continuous and also convex, implies that these functions hold inequality (2.9) for each fix p, i.e., (2.10)

holds.

”(ii) ⇒ (i)” Since f : [α,β] → R be a convex function, f ∈ C2 ([α,β]) and (ii) holds. Then the representation

of the function f in the form (2.1), (2.6), (2.7) and (2.8) for the functions Gc, c = 1, 2, 3, 4 implies that

for all s ∈ [α,β],
m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl) (2.11)

=
∫β
α

(
∑m
l=1 wlGc (xl,s) −

∑m
l=1 wlGc (yl,s)) f

′′(s)ds, c = 1, 2, 3, 4.

(2.12)

Since f is a convex function, then f
′′
(x) ≥ 0 for all x ∈ [α,β]. So, if for every s ∈ [α,β] the inequality (2.10)

holds for each c = 1, 2, 3, 4, then it follows that for every convex function f : [α,β] → R, with f ∈ C2[α,β],

inequality (2.9) holds.

At the end, note that it is not necessary to demand the existence of the second derivative of the function f

( [12], p.172). The differentiability condition can be directly eliminated by using the fact that it is possible

to approximate uniformly a continuous convex functions by convex polynomials. �

We give some identities related to the generalizations of majorization inequality by using Peano’s repre-

sentation of Hermite’s polynomial and new Green functions:

Theorem 2.2. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and

f ∈ Cn([α,β]) and w = (w1, ...,wm), x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that xl, yl



Int. J. Anal. Appl. 16 (3) (2018) 382

∈ [α,β],wl ∈ R (l = 1, ...,m). Also let Hij,GH,n and Gc(c = 1, 2, 3, 4) be as defined in (1.10), (1.12) and
(2.2)-(2.5) respectively. Then we have the following identities for c = 1, 2, 3, 4,

m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl) =

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α)

+

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

+

∫ β
α

f(n)(s)

[∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
GH,n−2(t,s)dt

]
ds,

(2.13)

where the Peano’s kernel (Green’s function) is defined as

GH,n−2(t,s) =




l∑
j=1

kj∑
i=0

(aj−s)n−i−3
(n−i−3)! Hij(t); s ≤ t,

−
r∑

j=l+1

kj∑
i=0

(aj−s)n−i−3
(n−i−3)! Hij(t); s ≥ t,

(2.14)

for all al ≤ s ≤ al+1; l = 0, . . . ,r with a0 = α and ar+1 = β.

Proof. Fix c = 1, 2, 3, 4, evaluating the identities one by one (2.1), (2.6), (2.7) and (2.8) into majorization

difference, we get

m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl)

=

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α) +

∫ β
α

(
m∑
l=1

wlGc (xl, t) −
m∑
l=1

wlGc (yl, t)

)
f′′(t)dt.

(2.15)

By the Peano’s representation of Hermite’s interpolatinhg polynomial Theorem 1.4, f′′(t) can be expressed

as

f′′(t) =

r∑
j=1

kj∑
i=0

Hij(t)f
(i+2)(aj) +

∫ β
α

GH,n−2(t,s)f
(n)(s)ds. (2.16)

Using (2.16) in (2.15) we get

m∑
l=1

wl φ (xl) −
m∑
l=1

wl φ (yl) =

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α)

+

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

+

∫ β
α

(
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

)(∫ β
α

GH,n−2(t,s)f
(n)(s)ds

)
dt.

after applying Fubini’s theorm we get (2.13). �

Integral version of the above theorem can be stated as:



Int. J. Anal. Appl. 16 (3) (2018) 383

Theorem 2.3. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points,

f ∈ Cn([α,β]) and x,y : [a,b] → [α,β], w : [a,b] → R be continuous functions. Also let Hij,GH,n−2 and
Gc(c = 1, 2, 3, 4) be as defined in (1.10), (2.14) and (2.2)-(2.5) respectively. Then we have the following

identities for c = 1, 2, 3, 4,∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ =

(∫ b
a

w(τ)x(τ)dτ −
∫ b
a

w(τ)y(τ)dτ

)
f
′
(α)

+

∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

+

∫ β
α

f(n)(s)

(∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
GH,n−2(t,s)dt

)
ds.

(2.17)

Theorem 2.4. Let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, w = (w1, ...,wm),

x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that xl, yl ∈ [α,β],wl ∈ R (l = 1, ...,m) and Hij,
Gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively. Let f : [α,β] → R be n−convex and

m∑
l=1

wl (Gc(xl, t) −Gc(yl, t)) ≥ 0, t ∈ [α,β]. (2.18)

Consider the inequalities for c = 1, 2, 3, 4,

m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl) ≥

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α)

+

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt. (2.19)

(i) If kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.19) hold.

(ii) If kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in

(2.19) hold.

Proof. (i) Since the function f is n−convex, therefore without loss of generality we can assume that φ

is n−times differentiable and f(n) ≥ 0 see [13, p. 16 and p. 293]. Also the given condition is that kj

is odd for each j = 1, 2, ..,r implies that

ω(t) =

r∏
j=1

(t−aj)
kj+1 ≥ 0.

By using the first part of Lemma 1.1 we have that the Peano’s kernel GH,n−2(t,s) ≥ 0. Hence, we

can apply Theorem 2.2 to obtain (2.19).

(ii) If kr is even then (t−ar)kr+1 ≤ 0 for any t ∈ [α,β]. Also clearly (t−a1)k1+1 ≥ 0 for any t ∈ [α,β]

and
∏r−1
j=2 (t−aj)

kj+1 ≥ 0 for t ∈ [α,β] if kj is odd for each j = 2, ..,r − 1, therefore combining all

these we have ω(t) =
∏r
j=1(t−aj)

kj+1 ≤ 0 for any t ∈ [α,β] and by using the first part of Lemma 1.1

we have GH,n−2(t,s) ≤ 0. Hence, we can apply Theorem 2.2 to obtain reverse inequality in (2.19).



Int. J. Anal. Appl. 16 (3) (2018) 384

�

Integral version of the above theorem can be stated as:

Theorem 2.5. Let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be given points and x,y : [a,b] → [α,β],

w : [a,b] → R be continuous functions and Hij and Gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5)
respectively. Let f : [α,β] → R be n−convex and∫ b

a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ ≥ 0, t ∈ [α,β]. (2.20)

Consider the inequalities for c = 1, 2, 3, 4,∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ ≥

(∫ b
a

w(τ)x(τ)dτ −
∫ b
a

w(τ)y(τ)dτ

)
f
′
(α)

+

∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt.

(2.21)

(i) If kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.21) hold.

(ii) If kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in

(2.21) hold.

By using type (m,n−m) conditions we can give the following result.

Corollary 2.1. Let [α,β] be an interval and w = (w1, ...,wp), x = (x1, ...,xp) and y = (y1, ...,yp) be p-tuples

such that xl, yl ∈ [α,β],wl ∈ R (l = 1, ...,p). Let Gc(c = 1, 2, 3, 4) be the Green functions as defined in
(2.2)-(2.5) respectively and also τi,ηi be as defined in (1.14) and (1.15) respectively. Let f : [α,β] → R be

n−convex and the inequality (2.18) holds for p-tuples. Consider the inequalities for c = 1, 2, 3, 4,
p∑
l=1

wl φ (xl) −
p∑
l=1

wl φ (yl) ≥

(
p∑
l=1

wlxl −
p∑
l=1

wlyl

)
f
′
(α)

+

∫ β
α

[
p∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

](
m−1∑
i=0

τi(t)f
(i+2)(α) +

n−m−1∑
i=0

ηi(t)f
(i+2)(β)

)
dt.

(2.22)

(i) If n−m is even, then the inequalities for c = 1, 2, 3, 4, in (2.22) hold.

(ii) If n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.22) hold.

By using Two-point Taylor conditions we can give the following result.

Corollary 2.2. Let [α,β] be an interval, w = (w1, ...,wp), x = (x1, ...,xp) and y = (y1, ...,yp) be p-tuples

such that xl, yl ∈ [α,β],wl ∈ R (l = 1, ...,p) and Gc(c = 1, 2, 3, 4) be the green function as defined in (2.2)-
(2.5) respectively. Let f : [α,β] → R be n−convex and the inequality (2.18) holds for p-tuples. Consider the



Int. J. Anal. Appl. 16 (3) (2018) 385

inequalities for c = 1, 2, 3, 4,

p∑
l=1

wl f (xl) −
p∑
l=1

wl f (yl) ≥

(
p∑
l=1

wlxl −
p∑
l=1

wlyl

)
f
′
(α)

+

∫ β
α

[
p∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

][
m−1∑
i=0

m−1−i∑
k=0

(
m + k − 1

k

)[(t−α)i
i!

( t−β
α−β

)m( t−α
β −α

)k
f(i+2)(α)

+
(t−β)i

i!

( t−α
β −α

)m( t−β
α−β

)k
f(i+2)(β)

]]
dt. (2.23)

(i) If m is even, then the inequalities for c = 1, 2, 3, 4, in (2.23) hold.

(ii) If m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.23) hold.

Remark 2.2. Similarly we can give integral version of Corollaries 2.1,2.2.

The following generalization of classical majorization theorem (also known as Karamata’s inequality) is

valid.

Theorem 2.6. Let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, x = (x1, ...,xm) and

y = (y1, ...,ym) be m-tuples such that y ≺ x with xl, yl ∈ [α,β] (l = 1, ...,m). Let Hij and Gc(c = 1, 2, 3, 4)

be as defined in (1.10) and (2.2)-(2.5) respectively and also f : [α,β] → R be n−convex. Consider the

inequalities for c = 1, 2, 3, 4,

m∑
l=1

f (xl) −
m∑
l=1

f (yl) ≥
∫ β
α

[
m∑
l=1

(Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt. (2.24)

(i) If kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.24) hold.

(ii) If kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in

(2.24) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.24) hold and the function

F(.) =
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(.) is non negative ( non positive), then the right hand side of (2.24) will be non

negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.2) will hold.

Proof. (i) Since the function G is convex and y ≺ x therefore by Theorem 1.1, the inequalities for c = 1, 2, 3, 4,

in (2.18) hold for wl = 1. Hence by Theorem 2.4(i) the inequalities for c = 1, 2, 3, 4, in (2.24) hold. Also if

the function F is convex then by using F in (1.2) instead of f we get that the right hand side of (2.24) is non

negative for each c = 1, 2, 3, 4.

Similarly we can prove part (ii). �

In the following theorem we give generalization of Fuch’s majorization theorem.



Int. J. Anal. Appl. 16 (3) (2018) 386

Theorem 2.7. Let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, x = (x1, ...,xm)

and y = (y1, ...,ym) be decreasing m-tuples and w = (w1, ...,wm) be any m-tuple with xl, yl ∈ [α,β],wl ∈ R
(l = 1, ...,m) which satisfy (1.3) and (1.4). Let Hij and Gc(c = 1, 2, 3, 4) be as defined in (1.10) and

(2.2)-(2.5) respectively and also f : [α,β] → R be n−convex, then

m∑
l=1

wlf (xl) −
m∑
l=1

wlf (yl) ≥
∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt.

(2.25)

(i) If kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.25) hold.

(ii) If kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in

(2.25) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.25) hold and the function

F(.) =
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(.) is non negative (non positive), then the right hand side of (2.25) will be non

negative (non positive) for c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.5) will hold.

Proof. Similar to the proof of Theorem 2.6. �

In the following theorem we give generalized majorization integral inequality.

Theorem 2.8. Let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, and x,y : [a,b] →

[α,β] be decreasing and w : [a,b] → R be continuous functions such that (1.6) and (1.7) hold. Also let Hij
and Gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively and also f : [α,β] → R be n−convex

and consider the inequalities for c = 1, 2, 3, 4,∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ

≥
∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt.

(2.26)

(i) If kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.26) hold.

(ii) If kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in

(2.26) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.26) hold and the function

F(.) =
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(.) is non negative (non positive), then the right hand side of (2.26) will be non

negative (non positive) for c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.8) will hold.

By using type (m,n−m) conditions we can give generalization of majorization inequality for majorized

tuples:



Int. J. Anal. Appl. 16 (3) (2018) 387

Corollary 2.3. Let [α,β] be an interval, x = (x1, ...,xp) and y = (y1, ...,yp) be any p-tuple such that y

≺ x with xl,yl ∈ [α,β] (l = 1, ...,p). Let τi and ηi be as defined in (1.14) and (1.15) respectively. Let

Gc(c = 1, 2, 3, 4) be defined as in (2.2)-(2.5) respectively and also f : [α,β] → R be n−convex. Consider the

inequalities for c = 1, 2, 3, 4,

p∑
l=1

f (xl) −
p∑
l=1

f (yl)

≥
∫ β
α

[
p∑
l=1

(Gc(xl, t) −Gc(yl, t))

](
m−1∑
i=0

τi(t)f
(i+2)(α) +

n−m−1∑
i=0

ηi(t)f
(i+2)(β)

)
dt.

(2.27)

(i) If n−m is even, then the inequalities for c = 1, 2, 3, 4 in (2.27) hold.

(ii) If n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.27) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.27) hold and the function

F(.) =
∑m−1
i=0 f

(i+2)(α)τi(.) +
∑n−m−1
i=0 f

(i+2)(β)ηi(.) is non negative (non positive), then the right hand side

of (2.27) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality)

in (1.2) will hold.

By using Two-point Taylor conditions we can give generalization of majorization inequality for majorized

tuples:

Corollary 2.4. Let [α,β] be an interval and x = (x1, ...,xp), y = (y1, ...,yp) be decreasing p-tuples such

that y ≺ x with xl,yl ∈ [α,β] (l = 1, ...,p). Let f : [α,β] → R be n−convex. Consider the inequalities for

c = 1, 2, 3, 4,
p∑
l=1

f (xl) −
p∑
l=1

f (yl) ≥
∫ β
α

[
p∑
l=1

(Gc(xl, t) −Gc(yl, t))

]
F(t)dt, (2.28)

where F(t) =
m−1∑
i=0

m−1−i∑
k=0

(
m + k − 1

k

)[
(t−α)i

i!

( t−β
α−β

)m( t−α
β −α

)k
f(i+2)(α)

+
(t−β)i

i!

( t−α
β −α

)m( t−β
α−β

)k
f(i+2)(β)

]
.

(i) If m is even, then the inequalities for c = 1, 2, 3, 4, in (2.28) hold.

(ii) If m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.28) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.28) hold and the function F(.) is non negative

(non positive), then the right hand side of (2.28) will be non negative (non positive) for each c = 1, 2, 3, 4,

that is the inequality (reverse inequality) in (1.2) will hold.

By using type (m,n−m) conditions we can give the following weighted majorization inequality.



Int. J. Anal. Appl. 16 (3) (2018) 388

Corollary 2.5. Let [α,β] be an interval and x = (x1, ...,xp) and y = (y1, ...,yp) be decreasing p-tuples and

w = (w1, ...,wp) be any p-tuple such that xl,yl ∈ [α,β],wl ∈ R (l = 1, ...,p) which satisfy (1.3) and (1.4).
Let τi and ηi be as defined in (1.14) and (1.15) respectively and let f : [α,β] → R be n−convex. Consider

the inequalities for c = 1, 2, 3, 4,

p∑
l=1

wl f (xl) −
p∑
l=1

wl f (yl)

≥
∫ β
α

[
p∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

](
m−1∑
i=0

τi(t)f
(i+2)(α) +

n−m−1∑
i=0

ηi(t)f
(i+2)(β)

)
dt.

(2.29)

(i) If n−m is even, then the inequalities for c = 1, 2, 3, 4, in (2.29) hold.

(ii) If n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.29) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.29) hold and the function

F(.) =
∑m−1
i=0 f

(i+2)(α)τi(.) +
∑n−m−1
i=0 f

(i+2)(β)ηi(.) is non negative (non positive), then the right hand side

of (2.29) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality)

in (1.5) will hold.

By using Two-point Taylor conditions we can give the following weighted majorization inequality.

Corollary 2.6. Let [α,β] be an interval and x = (x1, ...,xp), y = (y1, ...,yp) be decreasing p-tuples such

that xl, yl ∈ [α,β],wl ∈ R (l = 1, ...,p) which satisfy (1.3) and (1.4) and let f : [α,β] → R be n−convex.
Consider the inequalities for c = 1, 2, 3, 4,

p∑
l=1

wl f (xl) −
p∑
l=1

wl f (yl) ≥
∫ β
α

[
p∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
F(t)dt, (2.30)

where F(t) =
m−1∑
i=0

m−1−i∑
k=0

(
m + k − 1

k

)[
(t−α)i

i!

( t−β
α−β

)m( t−α
β −α

)k
f(i+2)(α)

+
(t−β)i

i!

( t−α
β −α

)m( t−β
α−β

)k
f(i+2)(β)

]
.

(i) If m is even, then the inequalities for c = 1, 2, 3, 4, in (2.30) hold.

(ii) If m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.30) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.30) hold and the function F(.) is non negative

(non positive), then the right hand side of (2.30) will be non negative (non positive) for each c = 1, 2, 3, 4,

that is the inequality (reverse inequality) in (1.5) will hold.

The integral version of the above Corollaries can be stated as:



Int. J. Anal. Appl. 16 (3) (2018) 389

Corollary 2.7. Let [α,β] be an interval and x,y : [a,b] → [α,β] be decreasing and w : [a,b] → R be
continuous function such that (1.6), (1.7) hold. Let τi and ηi be as defined in (1.14) and (1.15) respectively

and f : [α,β] → R be n−convex. Consider the inequalities for c = 1, 2, 3, 4,∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ

≥
∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

](
m−1∑
i=0

τi(t)f
(i+2)(α) +

n−m−1∑
i=0

ηi(t)f
(i+2)(β)

)
dt.

(2.31)

(i) If n−m is even, then the inequalities for c = 1, 2, 3, 4, in (2.31) hold.

(ii) If n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.31) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.31) holds and the function

F(.) =
∑m−1
i=0 f

(i+2)(α)τi(.) +
∑n−m−1
i=0 f

(i+2)(β)ηi(.) is non negative (non positive), then the right hand side

of (2.31) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality)

in (1.8) will hold.

Corollary 2.8. Let [α,β] be an interval and x,y : [a,b] → [α,β] be decreasing and w : [a,b] → R be
continuous functions such that (1.6) and (1.7) hold. Let f : [α,β] → R be n−convex. Consider the inequalities

for c = 1, 2, 3, 4,∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ ≥
∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
F(t)dt, (2.32)

where F(t) =
m−1∑
i=0

m−1−i∑
k=0

(
m + k − 1

k

)[
(t−α)i

i!

( t−β
α−β

)m( t−α
β −α

)k
f(i+2)(α)

+
(t−β)i

i!

( t−α
β −α

)m( t−β
α−β

)k
f(i+2)(β)

]
.

(i) If m is even, then the inequalities for c = 1, 2, 3, 4, in (2.32) hold.

(ii) If m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.32) hold.

If the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.32) hold and the function F(.) is non negative

(non positive), then the right hand side of (2.32) will be non negative (non positive) for each c = 1, 2, 3, 4,

that is the inequality (reverse inequality) in (1.8) will hold.

3. Upper Bounds for obtained generalized identities

For two Lebesgue integrable functions f,h : [α,β] → R we consider the Čebyšev functional

Λ(f,h) =
1

β −α

∫ β
α

f(t)h(t)dt−
1

β −α

∫ β
α

f(t)dt ·
1

β −α

∫ β
α

h(t)dt.

In [6] the authors proved the following theorems:



Int. J. Anal. Appl. 16 (3) (2018) 390

Theorem 3.1. Let f : [α,β] → R be a Lebesgue integrable function and h : [α,β] → R be an absolutely
continuous function with (·−α)(β −·)[h′]2 ∈ L[α,β]. Then we have the inequality

|Λ(f,h)| ≤
1
√

2
[Λ(f,f)]

1
2

1
√
β −α

(∫ β
α

(x−α)(β −x)[h′(x)]2dx

)1
2

. (3.1)

The constant 1√
2

in (3.1) is the best possible.

Theorem 3.2. Assume that h : [α,β] → R is monotonic nondecreasing on [α,β] and f : [α,β] → R is
absolutely continuous with f′ ∈ L∞[α,β]. Then we have the inequality

|Λ(f,h)| ≤
1

2(β −α)
‖f′‖∞

∫ β
α

(x−α)(β −x)dh(x). (3.2)

The constant 1
2

in (3.2) is the best possible.

In this section, we give the upper bounds like Grüss-type and Ostrowski-type for our generalized results.

For m-tuples w = (w1, ...,wm), x = (x1, ...,xm) and y = (y1, ...,ym) with xl, yl ∈ [α,β],wl ∈ R (l =
1, ...,m) and the Green functions Gc(c = 1, 2, 3, 4) and GH,n−2 be as defined in (2.2)-(2.5) and (2.14)

respectively, denote

L(s) =

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
GH,n−2(t,s)dt, s ∈ [α,β], (3.3)

for c = 1, 2, 3, 4, similarly for continuous functions x,y : [a,b] → [α,β], w : [a,b] → R and the Green function
Gc(c = 1, 2, 3, 4) and GH,n−2 be as defined in (2.2)-(2.5) and (2.14) respectively, denote

J(s) =

∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
GH,n−2(t,s)dt, s ∈ [α,β], (3.4)

for c = 1, 2, 3, 4.

Consider the Čebyšev functionals Λ(L,L), Λ(J,J) are given by:

Λ(L,L) =
1

β −α

∫ β
α

L2(s)ds−

(
1

β −α

∫ β
α

L(s)ds

)2
, (3.5)

Λ(J,J) =
1

β −α

∫ β
α

J2(s)ds−

(
1

β −α

∫ β
α

J(s)ds

)2
. (3.6)

Theorem 3.3. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points,

and f ∈ Cn([α,β]) such that (· − α)(β − ·)[f(n+1)]2 ∈ L[α,β] and w = (w1, ...,wm), x = (x1, ...,xm) and

y = (y1, ...,ym) be m-tuples such that xl, yl ∈ [α,β],wl ∈ R (l = 1, ...,m). Also let Hij be the fundamental



Int. J. Anal. Appl. 16 (3) (2018) 391

polynomials of the Hermite basis and the functions Gc(c = 1, 2, 3, 4) and L be defined by (2.2)-(2.5) and

(3.3) respectively. Then

m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl) =

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α)

+

r∑
j=1

kj∑
i=0

φ(i+2)(aj)

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
Hij(t)dt

+
f(n−1)(β) −f(n−1)(α)

β −α

∫ β
α

L(s)ds + REM(f; α,β), (3.7)

where the remainder REM(f; α,β) satisfies the estimation

|REM(f; α,β)| ≤
√
β −α
√

2
[Λ(L,L)]

1
2

∣∣∣∣∣
∫ β
α

(s−α)(β −s)[f(n+1)(s)]2ds

∣∣∣∣∣
1
2

. (3.8)

Proof. Comparing (3.7) and (2.13) we have

REM(f; α,β) = (β −α) Λ(L,f(n)).

Applying Theorem 3.1 on the functions L and f(n) we obtain (3.8). �

The integral version of the above theorem can be stated as:

Theorem 3.4. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points,

and f ∈ Cn([α,β]) such that (· − α)(β − ·)[f(n+1)]2 ∈ L[α,β] and x,y : [a,b] → [α,β], w : [a,b] → R be
continuous functions. Also let Hij be the fundamental polynomials of the Hermite basis and the functions

Gc(c = 1, 2, 3, 4) and J be defined by (2.2)-(2.5) and (3.4) respectively. Then∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ =

(∫ b
a

w(τ)x(τ)dτ −
∫ b
a

w(τ)y(τ)dτ

)
f
′
(α)

+

∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

+
f(n−1)(β) −f(n−1)(α)

β −α

∫ β
α

J(s)ds + ˜REM(f; α,β), (3.9)

where the remainder ˜REM(f; α,β) satisfies the estimation

∣∣∣ ˜REM(f; α,β)∣∣∣ ≤ √β −α√
2

[Λ(J,J)]
1
2

∣∣∣∣∣
∫ β
α

(s−α)(β −s)[f(n+1)(s)]2ds

∣∣∣∣∣
1
2

. (3.10)

Using Theorem 3.2 we obtain the following Grüss type inequalities.

Theorem 3.5. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and

f ∈ Cn([α,β]) such that f(n) is monotonic non decreasing on [α,β] and let L be defined by (3.3). Then the



Int. J. Anal. Appl. 16 (3) (2018) 392

representation (3.7) holds and the remainder REM(f; α,β) satisfies the bound

|REM(f; α,β)| ≤ ‖L′‖∞
{
f(n−1)(β) + f(n−1)(α)

2
−
f(n−2)(β) −f(n−2)(α)

β −α

}
. (3.11)

Proof. Since REM(f; α,β) = (β −α) Λ(L,f(n)), applying Theorem 3.2 on the functions L and f(n) we get

(3.11). �

Integral case of the above theorem can be given:

Theorem 3.6. Let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and

f ∈ Cn([α,β]) such that f(n) is monotonic non decreasing on [α,β] and let x,y : [a,b] → [α,β], w : [a,b] → R
be continuous functions and also Gc(c = 1, 2, 3, 4) and J be defined by (2.2)-(2.5) and(3.4) respectively. Then

we have the representation (3.9) and the remainder ˜REM(f; α,β) satisfies the bound

∣∣∣ ˜REM(f; α,β)∣∣∣ ≤‖J′‖∞{f(n−1)(β) + f(n−1)(α)
2

−
f(n−2)(β) −f(n−2)(α)

β −α

}
. (3.12)

We present the Ostrowski-type inequalities related to generalizations of majorization inequality.

Theorem 3.7. Suppose that all assumptions of Theorem 2.2 hold. Assume (u,v) is a pair of conjugate

exponents, that is 1 ≤ u,v ≤ ∞, 1/u + 1/v = 1. Let
∣∣f(n)∣∣u : [α,β] → R be an R-integrable function for

some n ∈ N. Then we have:

∣∣∣∣∣
m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl) −

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α)

−
∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

∣∣∣∣∣∣ ≤
∥∥∥f(n)∥∥∥

u
‖L‖v , (3.13)

where L is defined in (3.3).

The constant on the right-hand side of (3.13) is sharp for 1 < u ≤∞ and the best possible for u = 1.

Proof. By using (3.3) we have

L(t) =

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
GH,n−2(t,s)dt, for c = 1, 2, 3, 4.

Using the identity (2.13) and applying Hölder’s inequality we obtain∣∣∣∣∣
m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl)

−

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α) −

∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

∣∣∣∣∣∣
=

∣∣∣∣∣
∫ β
α

L(t)f(n)(t)dt

∣∣∣∣∣ ≤
∥∥∥f(n)∥∥∥

u

(∫ β
α

|L(t)|v dt

)1
v

.



Int. J. Anal. Appl. 16 (3) (2018) 393

For the proof of the sharpness of the constant
(∫β

α
|L(t)|v dt

)1
v

is analog to one in proof of Theorem 11

in [1]. �

Integral version of the above theorem can be given as:

Theorem 3.8. Suppose that all assumptions of Theorem 2.3 hold. Assume (u,v) is a pair of conjugate

exponents, that is 1 ≤ u,v ≤ ∞, 1/u + 1/v = 1. Let
∣∣f(n)∣∣u : [α,β] → R be an R-integrable function for

some n ∈ N. Then we have:

∣∣∣∣∣
∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ −

(∫ b
a

w(τ)(x(τ) −y(τ))dτ

)
f
′
(α)

−
∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt

∣∣∣∣∣∣
≤
∥∥∥f(n)∥∥∥

u
‖J‖v , (3.14)

where J is defined in (3.4).

The constant on the right-hand side of (3.14) is sharp for 1 < u ≤∞ and the best possible for u = 1.

4. n−exponential convexity and exponential convexity

We begin this section by giving some definitions and notions which are used frequently in the results. For

more details see e.g. [5], [9] and [14].

Definition 4.1. A function f : I → R is n-exponentially convex in the Jensen sense on I if
n∑

i,j=1

ξiξj f

(
xi + xj

2

)
≥ 0,

hold for all choices ξ1, . . . ,ξn ∈ R and all choices x1, . . . ,xn ∈ I. A function f : I → R is n-exponentially

convex if it is n-exponentially convex in the Jensen sense and continuous on I.

Definition 4.2. A function f : I → R is exponentially convex in the Jensen sense on I if it is n-exponentially

convex in the Jensen sense for all n ∈ N.

A function f : I → R is exponentially convex if it is exponentially convex in the Jensen sense and

continuous.

Proposition 4.1. If f : I → R is an n-exponentially convex in the Jensen sense, then the matrix
[
f
(
xi+xj

2

)]m
i,j=1

is a positive semi-definite matrix for all m ∈ N,m ≤ n. Particularly,

det

[
f

(
xi + xj

2

)]m
i,j=1

≥ 0,

for all m ∈ N, m = 1, 2, ...,n.



Int. J. Anal. Appl. 16 (3) (2018) 394

Remark 4.1. It is known that f : I → R+ is a log-convex in the Jensen sense if and only if

α2f(x) + 2αβf

(
x + y

2

)
+ β2f(y) ≥ 0,

holds for every α,β ∈ R and x,y ∈ I. It follows that a positive function is log-convex in the Jensen sense if

and only if it is 2-exponentially convex in the Jensen sense.

A positive function is log-convex if and only if it is 2-exponentially convex.

Motivated by inequalities (2.19) and (2.21), under the assumptions of Theorems 2.4 and 2.5 we define the

following linear functionals:

H1(f) =
m∑
l=1

wl f (xl) −
m∑
l=1

wl f (yl) −

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
f
′
(α)

−
∫ β
α

[
m∑
l=1

wl (Gc(xl, t) −Gc(yl, t))

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt, c = 1, 2, 3, 4,

(4.1)

and

H2(f) =
∫ b
a

w(τ)f(x(τ))dτ −
∫ b
a

w(τ)f(y(τ))dτ −

(∫ b
a

w(τ)(x(τ) −y(τ))dτ

)
f
′
(α)

−
∫ β
α

[∫ b
a

w(τ) (Gc(x(τ), t) −Gc(y(τ), t)) dτ

]
r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt, c = 1, 2, 3, 4.

(4.2)

Remark 4.2. Under the assumptions of Theorems 2.4 and 2.5, it holds Hi(f) ≥ 0, i = 1, 2, for all n−convex

functions f.

Lagrange and Cauchy type mean value theorems related to defined functionals are given in the following

theorems.

Theorem 4.1. Let f : [α,β] → R be such that f ∈ Cn[α,β]. If the inequalities in (2.18) (i = 1) and (2.20)
(i = 2) hold, then there exist ξi ∈ [α,β] such that

Hi(f) = f(n)(ξi)Hi(ϕ), i = 1, 2, (4.3)

where ϕ(x) = x
n

n!
and Hi, i = 1, 2 are defined by (4.1) and(4.2).

Proof. Similar to the proof of Theorem 4.1 in [10]. �

Theorem 4.2. Let f,g : [α,β] → R be such that f,g ∈ Cn[α,β]. If the inequalities in (2.18) (i = 1), (2.20)
(i = 2), hold, then there exist ξi ∈ [α,β] such that

Hi(f)
Hi(g)

=
f(n)(ξi)

g(n)(ξi)
, i = 1, 2, (4.4)

provided that the denominators are non-zero and Hi, i = 1, 2, are defined by (4.1) and(4.2).



Int. J. Anal. Appl. 16 (3) (2018) 395

Proof. Similar to the proof of Theorem 4.2 in [10]. �

Now we will produce n−exponentially and exponentially convex functions applying defined functionals.

We use an idea from [14]. In the sequel J will be interval in R.

Theorem 4.3. Let Ω = {ft : t ∈ J}, where J is an interval in R, be a family of functions defined on
an interval [α,β] such that the function t 7→ [x0, . . . ,xn; ft] is n−exponentially convex in the Jensen sense

on J for every (n + 1) mutually different points x0, . . . ,xn ∈ [α,β]. Then for the linear functionals Hi(ft)

(i = 1, 2) as defined by (4.1) and (4.2), the following statements hold:

(i) The function t → Hi(ft) is n-exponentially convex in the Jensen sense on J and the matrix [Hi(ftj+tl
2

)]mj,l=1

is a positive semi-definite for all m ∈ N,m ≤ n, t1, .., tm ∈ J. Particularly,

det[Hi(ftj+tl
2

)]mj,l=1 ≥ 0 for all m ∈ N, m = 1, 2, ...,n.

(ii) If the function t → Hi(ft) is continuous on J, then it is n-exponentially convex on J.

Proof. The proof is similar to the proof of Theorem 23 in [2]. �

The following corollary is an immediate consequence of the above theorem.

Corollary 4.1. Let Ω = {ft : t ∈ J}, where J is an interval in R, be a family of functions defined on an
interval [α,β] such that the function t 7→ [x0, . . . ,xn; ft] is exponentially convex in the Jensen sense on J for

every (n + 1) mutually different points x0, . . . ,xn ∈ [α,β]. Then for the linear functionals Hi(ft) (i = 1, 2)

as defined by (4.1) and (4.2), the following statements hold:

(i) The function t → Hi(ft) is exponentially convex in the Jensen sense on J and the matrix [Hi(ftj+tl
2

)]mj,l=1

is a positive semi-definite for all m ∈ N,m ≤ n, t1, .., tm ∈ J. Particularly,

det[Hi(ftj+tl
2

)]mj,l=1 ≥ 0 for all m ∈ N, m = 1, 2, ...,n.

(ii) If the function t → Hi(ft) is continuous on J, then it is exponentially convex on J.

Corollary 4.2. Let Ω = {ft : t ∈ J}, where J is an interval in R, be a family of functions defined on an
interval [α,β] such that the function t 7→ [x0, . . . ,xn; ft] is 2-exponentially convex in the Jensen sense on J

for every (n + 1) mutually different points x0, . . . ,xn ∈ [α,β]. Let Hi, i = 1, 2 be linear functionals defined

by (4.1) and (4.2). Then the following statements hold:

(i) If the function t 7→ Hi(ft) is continuous on J, then it is 2-exponentially convex function on J. If

t 7→ Hi(ft) is additionally strictly positive, then it is also log-convex on J. Furthermore, the following

inequality holds true:

[Hi(fs)]t−r ≤ [Hi(fr)]
t−s

[Hi(ft)]
s−r

, i = 1, 2,



Int. J. Anal. Appl. 16 (3) (2018) 396

for every choice r,s,t ∈ J, such that r < s < t.

(ii) If the function t 7→ Hi(ft) is strictly positive and differentiable on J, then for every p,q,u,v ∈ J,

such that p ≤ u and q ≤ v, we have

µp,q(Hi, Ω) ≤ µu,v(Hi, Ω), (4.5)

where

µp,q(Hi, Ω) =



(

Hi(fp)
Hi(fq)

) 1
p−q

, p 6= q,

exp

(
d
dp

Hi(fp)
Hi(fp)

)
, p = q,

(4.6)

for fp,fq ∈ Ω.

Proof. The proof is similar to the proof of Corollary 2 in [2]. �

Remark 4.3. Note that the results from Theorem 4.3, Corollary 4.1 and Corollary 4.2 still hold when two

of the points x0, ...,xl ∈ [α,β] coincide, say x1 = x0, for a family of differentiable functions ft such that

the function t 7→ [x0, ...,xl; ft] is an n-exponentially convex in the Jensen sense (exponentially convex in

the Jensen sense, log-convex in the Jensen sense), and furthermore, they still hold when all (l + 1) points

coincide for a family of l differentiable functions with the same property. The proofs are obtained by suitable

characterization of convexity.

5. Applications

In this section, we give some applications of our generalized results about the upper bounds as well as

exponential convex functions.

Firstly, we consider some related analytical inequalities by using our generalized results of upper bounds.

Example 5.1. By using Ostrowski-type inequality (3.13) for n = 4 as an upper bound of our generalized

results,

• let f(x) = ex, x ∈ R, then

0 ≤|
m∑
l=1

wle
xl −

m∑
l=1

wle
yl −

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
eα −GHc |≤

(euβ −euα)
1
u

u
1
u

‖ L ‖v,

• let f(x) = xr, [0,∞) for r > 3, then

0 ≤|
m∑
l=1

wlx
r
l −

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
rαr−1 −GHc |

≤
r(r − 1)(r − 2)(r − 3)

(u(r − 4) + 1)
1
u

(
βu(r−4)+1 −αu(r−4)+1

) 1
u

‖ L ‖v,



Int. J. Anal. Appl. 16 (3) (2018) 397

• let f(x) = x log x, x ∈ (0,∞), then

0 ≤|
m∑
l=1

wlxl log xl −
m∑
l=1

wlyl log yl −

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
(log α + 1) −GHc |

≤
2

(1 − 3u)
1
u

(
β1−3u −α1−3u

) 1
u ‖ L ‖v,

• let f(x) = − log x, x ∈ (0,∞), then

0 ≤|
m∑
l=1

wl log yl −
m∑
l=1

wl log xl +

(
m∑
l=1

wlxl −
m∑
l=1

wlyl

)
1

α
−GHc |

≤
6

(1 − 4u)
1
u

(
β1−4u −α1−4u

) 1
u ‖ L ‖v,

where, GHc =
∫β
α

[
∑m
l=1 wl (Gc(xl, t) −Gc(yl, t))]

r∑
j=1

kj∑
i=0

f(i+2)(aj)Hij(t)dt, (c = 1, 2, 3, 4), and L(s) =∫β
α

[
∑m
l=1 wl (Gc(xl, t) −Gc(yl, t))] GH,2(t,s)dt.

We can also give the particular cases of above results for u = 1 and v = ∞.

Now, we construct exponentially convex function by using family of convex functions defined on (0,∞):

Example 5.2. Let

E1 = {θv : (0,∞) → (0,∞) : v ∈ R}

be a family of continuous convex functions defined by

θv(x) =




xevx

v2
, v 6= 0;

x3

2
, v = 0.

We have v 7→
(
θv(x)
x

)′′
(t ∈ R) is exponentially convex for every fixed x ∈ R. Using analogous arguing as

in the proof of Theorem 4.3 we also have that v 7→ θv[z0, ...,zt] is exponentially convex (and so exponentially

convex in the Jensen sense). Using Corollary 4.1 we conclude that v 7→ Hi(θv) is exponentially convex in

the Jensen sense. It is easy to verify that this mapping is continuous (although mapping v 7→ θv is not

continuous for v = 0), so it is exponentially convex.

For this family of functions, µv,q (E1,Hi) from (4.6), becomes

µp,q(Hi,E1) =




(
Hi(θp)
Hi(θq)

) 1
p−q

, p 6= q,

exp
(

Hi(id·θp)
Hi(θp)

− n
p

)
, p = q 6= 0,

exp
(

1
n+1

Hi(id·φ0)
Hi(θ0)

)
, p = q = 0,

where id is the identity function.



Int. J. Anal. Appl. 16 (3) (2018) 398

Now using (4.5), µp,q is monotone function in parameters p and q.

We observe here that

(
d2θp

dx2

d2θq

dx2

) 1
p−q

(lnx) = x so using Theorem 4.2 it follows that

Mp,q(E1,Hi) = lnµp,q(E1,Hi),

satisfies

α ≤ Mp,q(E1,Hi) ≤ β, i = 1, 2.

This shows that Mp,q(E1,Hi) is mean. Because of the above inequality (4.5), this mean is also monotonic.

Remark 5.1. We can construct other examples for exponentially convex functions as Example 2 for the

families of continuous convex functions:

•

E2 = {µt : (0,∞) → R : t ∈ R}

where,

µt(x) =




xt+1

t(t−1), t 6= 0, 1;

−x log x, t = 0;

x2 log x, t = 1.

•

E3 = {χt : (0,∞) → (0,∞) : t ∈ (0,∞)}

where,

χt(x) =




xt−x

log2 t
, t 6= 1;

x3

2
, t = 1.

•

E4 = {δt : (0,∞) → (0,∞) : t ∈ (0,∞)}

where,

δt(x) :=
xe−x

√
t

t
.

Conflict of Interests:

The authors declare that there is no conflict of interests.

Acknowledgment:

The publication was supported by the Ministry of Education and Science of the Russian Federation (the



Int. J. Anal. Appl. 16 (3) (2018) 399

Agreement number No. 02.a03.21.0008.) This publication is partially supported by Royal Commission for

Jubail and Yanbu, Kingdom of Saudi Arabia.

References

[1] R. P. Agarwal, S. Ivelić Bradanović and J. Pečarić, Generalizations of Sherman’s inequality by Lidstone’s interpolating

polynomial, J. Inequal. Appl., 2016 (2016), Art. ID 6.

[2] M. Adil Khan, N. Latif and J. Pečarić, Generalization of majorization theorem, J. Math. Inequal., 9(3) (2015), 847-872.

[3] R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Academic

Publishers, Dordrecht/ Boston/ London, 1993.

[4] P. R. Beesack, On the Greens function of an N-point boundary value problem, Pacific J. Math. 12 (1962), 801-812. Kluwer

Academic Publishers, Dordrecht / Boston / London, 1993.

[5] S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1–66.

[6] P. Cerone and S. S. Dragomir, Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math.

Inequal. 8(1) (2014), 159–170.

[7] P. J. Davis, Interpolation and Approximation, Blaisedell Publishing Co., Boston, 1961.

[8] L. Fuchs, A new proof of an inequality of Hardy-Littlewood-Polya, Mat. Tidsskr, B (1947), 53-54.

[9] J. Jakšetić and J. Pečarić, Exponential convexity method, J. Convex Anal. 20(2013), no. 1, 181-197.

[10] J. Jakšetić, J. Pečarić and A. Perušić, Steffensen inequality, higher order convexity and exponential convexity, Rend. Circ.

Mat. Palermo 63 (1) (2014), 109–127.

[11] N. Mahmood, R. P. Agarwal, S. I. Butt and J. Pečarić, New Generalization of Popoviciu type inequalities via new Green

functions and Montgomery identity, J. Inequal. Appl., 2017 (2017), Art. ID 108.

[12] A. W. Marshall, I. Olkin and Barry C. Arnold, Inequalities: Theory of Majorization and Its Applications (Second Edition),

Springer Series in Statistics, New York 2011.

[13] J. Pečarić, F. Proschan and Y. L. Tong, Convex functions, Partial Orderings and Statistical Applications, Academic Press,

New York, 1992.

[14] J. Pečarić and J. Perić, Improvements of the Giaccardi and the Petrović inequality and related results, An. Univ. Craiova

Ser. Mat. Inform., 39(1) (2012), 65–75.

[15] J. Pečarić, On some inequalities for functions with nondecreasing increments, J. Math. Anal. Appl., 98 (1984), 188-197.

[16] A. Yu. Levin, Some problems bearing on the oscillation of solutions of linear differential equations, Soviet Math. Dokl.,

4(1963), 121-124.


	1. Introduction and Preliminaries
	2. Main Results via Peano's representation and new Green functions
	3. Upper Bounds for obtained generalized identities
	4. n-exponential convexity and exponential convexity
	5. Applications
	References