EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5694
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analysis of Hardy-type Inequalities Involving Green
Functions and Taylor’s Polynomial Approximation

Anjum Mustafa Khan Abbasi1,∗, Matloob Anwar1

1 Department of Mathematics School of Natural Sciences, National University of Sciences
and Technology, Islamabad, Pakistan

Abstract. Herein, we prove the Hardy-type inequalities using the two-point right focal problem’s
Green functions, which are convex and continuous concerning both variables. Along with the Green
functions, Taylor’s polynomial and n−convex functions are also considered. In addition, we find
the bounds on the remainder using Čebyšev functional in the presence of obtained results in the
form of Hardy-type inequalities. Then, we discuss Grüss-type inequalities that enable us to find
the bounds on remainders and then Ostrowski-type inequalities are discussed. In the later part of
this study, we discuss some results related to the mean value theorem and n-exponential convexity.

2020 Mathematics Subject Classifications: 26D10, 26D15, 26D25, 26D30

Key Words and Phrases: Hardy-type Inequalities, Convex functions, n-Exponential Convex
functions, Green function, Taylor interpolating polynomial, Čebyšev functional, Grüss-type in-
equalities, Ostrowski-type inequalities, Bounds on remainders

1. Introduction

A remarkable development in the theory of Green function was introduced by Green
[8] that was almost seminal and later on theory related to Green function plays an im-
portant role in dealing with differential equations qualitatively and quantitatively [4],[13].
Mathematicians used Green functions in many other directions of mathematics, other than
differential equations. In resent past, it has been used diversely in the mathematical in-
equalities along with the polynomial interpolation. Like in [2] different aspects have been
extensively discussed. Then further generalizations have been made by many researchers
like K. K .Himmelreich et al. [10] discuss the Hardy-type inequalities by tkaing Green
function into account along with Montgomery identity. D. Pokaz [20] studied the Hardy-
type inequalities via Green functions, n−convex functions and polynomial interpolation of
Abel Gontschorf and find the bounds on the remainder obtained from the assumptions un-
der consideration. A. Rasheed et al. [22] investigate the Levinson-type inequalities using

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5694

Email addresses: aabbasi.phdmath19sns@student.nust.edu.pk (A. M. K. Abbasi),
manwar@sns.nust.edu.pk (M. Anwar)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 2 of 18

Green functions of the two-point right focal problems. Muhammad Adeel et al. investigate
Levisnson type inequalities in [1] by using the Green function and Lidstone’s polynomial
and then he gave the application of these inequalities to the estimate the f−divergence and
Shannon entropy. In [7] some real-world problems are discussed along with the Hermite-
Hadamard inequality and Green functions. Although this inequality is vastly studied in
many other ways like it’s generalizations so-called Levinson’s inequality or involvement of
fractional calculus to study this inequality but we analyze the aforementioned inequality
for upcoming Green functions and Taylor’s polynomial.

Due to the importance of the Hardy-type inequalities to find the prior estimation to
the solution of partial differential equations, approximation of a function to the basis
of functions or the polynomials, it got the attention of mathematicians. Some nominal
works and discussion are made in [2], [22], [9], [23] and [12]. Also, there is scope to extend
this work in the fractional calculus side using different operators discussed in [6], [7] and
[21] in the presence of Čebyšev functional in fractional sense accordingly. For example,
while discussing the aforementioned inequality in conformable fractional operator we use
Čebyšev functional given in [24], but in the present study we are considering the ordinary
case. For further details it is plausibe to mention some important and nominal work that
generalize this inequaltiy by using polynomial interpolation and Green function. Kristina
Krulić Himmelreich [15] discuss Hardy-type inequalities via Taylor’s polynomial, D Pokaz
in [20] use Abel-Gontscharoff interpolation and the Green functions as well to generalize
aforementioned inequality and Kristina Krulić Himmelreich et al. [14] generalize Hardy-
type inequalities with the Hermite interpolating polynomials.

We observe that the Hardy-type inequalities are not yet studied in the presence of the
Green functions presented in Lemma 1 of upcoming section. Herein, we are interested in
generalizing Hardy-type inequalities via Taylor’s polynomial and the two-point right focal
Green function, which is discussed in section 3. After that, we find the Grüss-type and
Ostowski-type bounds that are given in section 4. The final section 5 involves some results
regarding the n-exponential convexity and utility of our previous results, especially the
functional obtained from Theorem 5.

2. Preliminaries

Initiating by considering the following Lemma involving the Green functions that are
3-convex. Also, these Green functions are continuous and convex with respect to the
involved variables and given in [22] as;

Lemma 1. Assuming the real valued function F defined on T = [a1, a2] and F is thrice
differentiable therein. Let Gα(α = {1, 2, 3, 4}) be the two-point right-focal problem type
Green function. Then

F(ϖ) = F(a1) + (ϖ − a1)F ′(a2) + (ϖ − a1)(ϖ − a2)F ′′(a1)−
(ϖ − a1)

2

2
F ′′(a2)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 3 of 18

+

∫ a2

a1

G1(ϖ, τ)F ′′′(τ)dτ, (1)

F(ϖ) = F(a2) + (ϖ − a2)F ′(a1) + (ϖ − a1)(ϖ − a2)F ′′(a2)−
(ϖ − a2)

2

2
F ′′(a1)

−
∫ a2

a1

G2(ϖ, τ)F ′′′(τ)dτ, (2)

F(ϖ) = F(a2) + (ϖ − a1)F ′(a1) + (a2 − a1)F ′(a2)−
[
(ϖ − a1)

2

2
+ (ϖ − a1)(a− b)

]
F ′′(a1)

+

[
(a2 − a1)

2

2
+ (ϖ − a1)(ϖ − a2)

]
F ′′(a2)−

∫ a2

a1

G3(ϖ, τ)F ′′′(τ)dτ (3)

and

F(ϖ) = F(a2) + (a2 − a1)F ′(a1) + (b−ϖ)F ′(a2) +

[
(a2 − a1)

2

2
+ (ϖ − a2)(ϖ − a1)

]
F ′′(a1)

−
[
(ϖ − a2)

2

2
+ (ϖ − a2)(a2 − a1)

]
F ′′(a2) +

∫ a2

a1

G4(ϖ, τ)F ′′′(τ)dτ. (4)

where the Green functions are given as

G1(ϖ, τ) =

{
(τ−a1)2

2 + (ϖ − a1)(ϖ − a2) a ≤ τ ≤ ϖ

(ϖ − a1)(τ − a2) +
(τ−a1)2

2 ϖ ≤ τ ≤ b
, (5)

G2(ϖ, τ) =

{
(τ−a2)2

2 + (τ − a1)(ϖ − a2) a ≤ τ ≤ ϖ

(ϖ − a1)(ϖ − a2) +
(τ−a2)2

2 ϖ ≤ τ ≤ b
, (6)

G3(ϖ, τ) =

{
(τ−a1)2

2 + (ϖ − a1)(τ − a2) a ≤ τ ≤ ϖ

(ϖ − a1)(ϖ − a2) +
(τ−a1)2

2 ϖ ≤ τ ≤ b
, (7)

G4(ϖ, τ) =

{
(τ−a2)2

2 + (ϖ − a1)(ϖ − a2) a ≤ τ ≤ ϖ

(ϖ − a2)(τ − a1) +
(ϖ−a2)2

2 ϖ ≤ τ ≤ b.
(8)

The Taylor’s formula for the function F : T = [a1, a2] → R at c ∈ T and F (n−1) have
the property of absolutely continuity, is;

F(ϖ) =
n−1∑
κ=0

f (κ)(ϖ)

κ!
(ϖ − c)κ +

1

n− 1

∫ ϖ

c
F (n)(τ)(ϖ − τ)m−1dτ. (9)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 4 of 18

Lemma 2. Assuming that F : T = [a1, a2] → R is such that F (n−1) is absolute continuous
and ϖ ∈ T . Then the Taylor’s formula at point a1 and a2 is given in [2] as;

F(ϖ) =
n−1∑
κ=0

F (κ)(a1)

κ!
(ϖ − a1)

κ +
1

(n− 1)!

∫ a2

a1

F (n)(τ)(ϖ − τ)n−1
+ dτ. (10)

where ∫ a2

a1

(ϖ − τ)n−1
+ dτ =

∫ ϖ

a1

(ϖ − τ)n−1dτ +

∫ a2

ϖ
0dτ,

and

F(ϖ) =
n−1∑
κ=0

F (κ)(a2)

κ!
(−1)κ(a2 −ϖ)κ − (−1)n−1

(n− 1)!

∫ a2

a1

F (n)(τ)(τ −ϖ)n−1
+ dτ. (11)

where ∫ a2

a1

(τ −ϖ)n−1
+ dτ =

∫ ϖ

a1

0dτ +

∫ a2

ϖ
(τ −ϖ)n−1dτ,

(τ −ϖ)+ =

{
τ −ϖ ϖ ≤ τ

0 ϖ > τ
. (12)

Definition 1. The n-th order divided difference of a real valued function f defined on
[a1, a2] at distinct points ϖ0, ..., ϖn ∈ [a1, a2] is defined for (i = 0, 1, ..., n) as

f [ϖi] = f(ϖi),

f [ϖ0, ..., ϖn] =
f [ϖ1, ..., ϖn]− f [ϖ0, ..., ϖn−1]

ϖn −ϖ0
.

Here the value obtained from f [ϖ0, ..., ϖn] is not depending on the order of points
ϖ0, ..., ϖn. In case when all points coincides then this definition will be extended to the
following form if f (j−1)(ϖ) exists

f [ϖ, ...j−times, ϖ] =
f (j−1)(ϖ)

(j − 1)!
. (13)

The British mathematician G H Hardy introduce an inequality so-called Hardy inequality
in [9] given as;∫ ∞

0

( 1

ϖ

∫ ϖ

0
f(τ)dτ

)γ
dϖ ≤

( γ

γ − 1

)γ ∫ ∞

0
fγ(ϖ)dϖ, γ > 1, (14)

where he took a non-negative function f with f ∈ Lγ(R+) and R+ = (0,∞) along with

sharp constant
(

γ
γ−1

)γ
. With the passage of time it has been generalized in many ways.



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 5 of 18

Like in 1964 N Levinson generalized Hardy inequality given [19], [18]. But [18], [16] and
[17] generalized the Hardy-type inequalities and operator. After some generalizations S.
Kaisjer et al. [12] gave some nominal concept of Hardy-type inequalities via convexity.
Here we throw a glance on some of their important results. For positive σ−finite measures,
the two measure spaces (

∑
1,Ω1, σ1) and (

∑
2,Ω2, σ2), we have Ak the operator given in

[12] as

Akf(ϖ) =
1

K(ϖ)

∫
Ω2

G(ϖ, τ)f(τ)dµ2(τ), (15)

where f is the measurable function and G : Ω1×Ω2 → R is nonnegative measurable kernel
and obeys the following inequality

0 < K(ϖ) =

∫
Ω2

G(ϖ, τ)dµ2(τ), ϖ ∈ Ω1. (16)

Furthermore S. Kaisjer et al. gave another useful result in [12] in the form of following
theorem.

Theorem 1. Assuming that G(ϖ,s)
K(ϖ) u(ϖ) is integrable locally on Ω1 for each fixed s ∈ Ω2,

where u be the weight function, G(ϖ, s) ≥ 0. Define v as

ν(s) =

∫
Ω1

G(ϖ, s)

K(ϖ)
u(ϖ)dµ1(ϖ) < ∞. (17)

If F is supposed to be the convex function on I ⊂ R which is open, then following relation
holds ∫

Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ) ≤
∫
Ω2

F(f(s))ν(s)dµ2(s). (18)

for measurable function f : Ω1 → R, with image of f is subset of I, and Ak is given in
(15).

Now we are discussing some of the existing results from literature which help us to find
some important bounds using Ostrowski and Grüss-type inequalities presented in section
4. Considering the following functional for two real valued Lebesgue integrable functions
f, g over an interval T so-called Čebyšev functional given by P. Cerone et al. in [5] as

F(f, g) =
1

a2 − a1

∫ a2

a1

f(η)g(η)dη − 1

a2 − a1

∫ a2

a1

f(η)dη.
1

a2 − a1

∫ a2

a1

g(η)dη. (19)

Next result is given in the same article [5] which is used to find the bounds of the remainder
of Hardy-type inequalities using Taylor’s polynimial and Green function.



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 6 of 18

Theorem 2. Let the functions f, g are as taken in (19) and uniformly continuous on
[a1, a2] as well. Additionally assuming (. − a), (b − .)(g′)2 ∈ L[a1, a2], then following
inequality holds

∣∣∣F(f, g)∣∣∣ ≤ 1√
2

(
F(f, f)

) 1
2 1√

a2 − a1

(∫ a2

a1

(η − a1)(a2 − η)(g′(η))2dη

)2

, (20)

where 1√
2
is the best possible approximation.

Also, the Grüss-type inequality was given in [5], which is given as follows;

Theorem 3. Let f is absolutely continuous on [a1, a2] and f ′ ∈ L∞[a1, a2] and g :
[a1, a2] → R is monotonic nondecreasing then following result holds∣∣∣F(f, g)∣∣∣ ≤ 1

2(a2 − a1)
||f ′||∞

∫ a2

a1

(η − a1)(a2 − η)(g′(η))2dg(η), (21)

where 1
2 is the best possible approximation.

The following definitions of n-exponential convexity from [11] is used in section 5 and
related results of exponential convexity are under consideration there.

Definition 2. A function F : T → R is n-exponentially convex in Jensen sense on T if

n∑
i,j

cicjF
(ζi + ζj

2

)
≥ 0, (22)

holds for all c1, c2, ..., cn ∈ R and all choices of ζ1, ..., ζn ∈ T .

A function F : T → R will be n-exponentially convex if it meets the criteria for
n-exponential convexity in Jensen sense and continuous on T .

Remark 1. The aforementioned definition ensures that the function which is 1-exponentially
convex in Jensen sense is indeed non-negative. Furthermore, an n-exponentially convex
function in Jensen sense is also l-exponentially convex in Jensen sense for each l ∈ N,
where l ≤ n.

Employing the definition of semi-definite matrices and some fundamental results from
linear algebra, J.Pečarić [11] propose the following result:

Proposition 1. If F is n-exponentially convex in Jensen sense, then the matrix[
F
(ζi + ζj

2

)]l
i,j=1

, ∀ l ∈ N, l ≤ n,

is positive semi definite. In particular following result holds for all such l

det
[
F
(ζi + ζj

2

)]l
i,j=1

≥ 0.



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 7 of 18

Definition 3. If a function F : T → R is n-exponentially convex in Jensen sense for all
n ∈ N then F is eponentially convex in Jensen sense. A continuous function F : T → R
that is eponentially convex in Jensen sense is exponentially convex.

Remark 2. It is easy to show that F : T → R is a log-convex in the Jensen sense if and
only if

a21F(ζ1) + 2a1a2F
(ζ1 + ζ2

2

)
+ a22F(ζ2) ≥ 0, (23)

holds for all a1, a2 ∈ R and ζ1, ζ2 ∈ T . Consequently, the function is 2-exponentially
convex in the Jensen sense if and only if it is log-convex in the Jensen sense.

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

3. Main Results

In this section, we present Hardy-type inequalities as a fundamental result, employ-
ing Green functions obtained from the two-point right focal problem in addition to this,
Taylor’s polynomial is also taken into account, after that, using these fundamental in-
equalities we analyze the Grüss-type and Ostrowski-type bounds. Starting with our first
result which is given as;

Theorem 4. Let F be defined on T such that F ′′′ exists and F (n−1) is absolutely contin-
uous therein, for n ∈ N. Considering two measure spaces (

∑
1,Ω1, σ1) and (

∑
2,Ω2, σ2)

with positive σ−finite measures and Ak and K are mentioned in (15) and (16) respectively.
Let Gα, {α = 1, 2, 3, 4} be defined in (5), (6), (7) and (8) respectively and u : Ω1 → R be
the weight function and v is given in (17), then:

• ∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

=

∫ a2

a1

JGα(f, τ)
[∑

κ

F (κ)(a1)

κ!
(τ − a1)

κ

+
1

(n− 4)!

∫ a2

a1

F (n)(η)(τ − η)n−4
+ dη

]
dτ, (24)

• ∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

=

∫ a2

a1

JGα(f, τ)
[∑

κ

(−1)κF (κ)(a2)

κ!
(a2 − τ)κ

−(−1)n−4

(n− 4)!

∫ a2

a1

F (n)(η)(τ − η)n−4
+ dη

]
dτ, (25)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 8 of 18

where

JGα(f, τ) =

∫
Ω2

Gα(f(s), τ)ν(s)dµ2(s)−
∫
Ω1

Gα(Akf(ϖ), τ)u(ϖ)dµ1(ϖ) (26)

and notation
∑

κ is used for
∑n−1

κ=3 throughout this paper.

Proof.

• First we consider α = 1 and from (1) and we can write∫
Ω2

F(f(s))ν(s)dµ2(s) =

∫
Ω2

[
F(a1) + (f(s)− a)F ′(a2) + (f(s)− a)(f(s)− b)F ′′(a1)

−(f(s)− a)2

2
F ′′(a2) +

∫ a2

a1

G1(f(s), τ)F ′′′(τ)dτ
]
ν(s)dµ2(s). (27)

Similarly, we can write

∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

=

∫
Ω2

[
F(a1) + (Akf(ϖ)− a)F ′(a2) + (Akf(ϖ)− a)(Akf(ϖ)− b)F ′′(a1)

−(Akf(ϖ)− a)2

2
F ′′(a2) +

∫ a2

a1

G1(Akf(ϖ), τ)F ′′′(τ)dτ
]
u(ϖ)dµ1(ϖ). (28)

Now subtracting (28) from (27) we obtain

∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

=

∫
Ω2

[ ∫ a2

a1

G1(f(s), τ)F ′′′(τ)dτ
]
ν(s)dµ2(s)

−
∫
Ω1

[ ∫ a2

a1

G1(Akf(ϖ), τ)F ′′′(τ)dτ
]
u(ϖ)dµ1(ϖ). (29)

The fact used in obtaining the equation (29) is as follows;∫
Ω2

ν(s)dµ2(s) =

∫
Ω2

∫
Ω1

k(ϖ, s)

K(ϖ)
u(ϖ)dµ1(ϖ)dµ2(s)

=

∫
Ω1

u(ϖ)

K(ϖ)

∫
Ω2

G(ϖ, s)dµ2(s)dµ1(ϖ)

=

∫
Ω1

u(ϖ)

K(ϖ)
K(ϖ)dµ1(ϖ) =

∫
Ω1

u(ϖ)dµ1(ϖ)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 9 of 18

and ∫
Ω2

Akf(ϖ)u(ϖ)dµ1(ϖ) =

∫
Ω1

[ 1

K(ϖ)

∫
Ω2

G(ϖ, s)f(s)dµ2(s)
]
u(ϖ)dµ1(ϖ)

=

∫
Ω2

f(s)

∫
Ω1

G(ϖ, s)

K(ϖ)
u(ϖ)dµ1(ϖ)dµ2(s)

=

∫
Ω2

f(s)ν(s)dµ2(s).

Now applying (10) in F ′′′ at point a we obtain

F ′′′(τ) =
∑
κ

F (κ)(a1)

κ!
(τ − a1)

j−3 +
1

(n− 4)!

∫ a2

a1

F (n)(η)(τ − η)n−4
+ dη. (30)

Using (30) in (29) we obtain for α = 1 as

∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

=

∫ a2

a1

[∫
Ω2

Gα(f(s), τ)ν(s)dµ2(s)−
∫
Ω1

Gα(Akf(ϖ), τ)u(ϖ)dµ1(ϖ)

]

×
∑
κ

F (κ)(a1)

κ!
(τ − a1)

κ−3dτ

+

∫ a2

a1

([∫
Ω2

Gα(f(s), τ)ν(s)dµ2(s)−
∫
Ω1

Gα(Akf(ϖ), τ)u(ϖ)dµ1(ϖ)

]

× 1

(n− 4)!

∫ a2

a1

F (n)(η)(τ − η)n−4
+ dη

)
dτ.

After giving the notation mentioned in (26) we arrived at the required result.

For α = 2, 3, 4 consider the left hand sides of (27) and (28), then make use of
(2), (3) and (4), after that subtracting the obtained results to get (29) then follow
the same steps as for α = 1 to obtain the required result.

• Proof is analogous to the first part, use the Taylor polynomial (11) and simplify we
get required.

Remark 3. As Gα are convex with respect to both variables, replacing the convex function
F(.) by Gα(., τ) ensures that relation in (18) remains valid, that is;∫

Ω1

Gα(Akf(ϖ), τ)u(ϖ)dµ1(ϖ) ≤
∫
Ω2

Gα(f(s), τ)ν(s)dµ2(s). (31)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 10 of 18

Theorem 5. Let all conditions of Theorem 4 holds and if F is n− convex on T ,
then;

• (i) ∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

≥
∫ a2

a1

JGα(f, τ)×
∑
κ

F (κ)(a1)

κ!
(τ − a1)

κdτ. (32)

• (ii) For any odd n > 4;∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

≥
∫ a2

a1

JGα(f, τ)×
∑
κ

(−1)κ
F (κ)(a2)

κ!
(a2 − τ)κdτ. (33)

Proof.

• As F is n−convex for all n ∈ N implies F (n) ≥ 0 for τ ∈ T . Also, (τ − η)n−4
+

is non-negative for all n ∈ N because τ ≥ η. By using aforementioned reasons in
equation (24) we obtain the required result.

• As F is n−convex this implies F (n) ≥ 0 on T . Also,

(−1)n−4(τ − η)n−4
+ =

{
0 τ ≤ η

(η − τ)n−4 τ ≥ η
, (34)

Since for τ ≥ η, so for any odd n ≥ 4, (η − τ)n−4 is negative and −(η − τ)n−4 will
be nonnegative. Thus last term in (25) is nonnegative. Hence we can write (33).

Theorem 6. Let the conditions of Theorem 4 hold entirely. Additionally, if F is n−convex
and the function;

(i)

L1(.) =
∑
κ

F (κ)(a1)

κ!

∫ a2

a1

Gα(., τ)(τ − a1)
κdτ, (35)

is convex on [a1, a2], then

∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ) ≤
∫
Ω2

F(f(s))ν(s)dµ2(s). (36)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 11 of 18

(ii)

L2(.) =
∑
κ

(−1)κF (κ)(a1)

κ!

∫ a2

a1

Gα(., τ)(τ − a1)
κdτ, (37)

is convex on [a1, a2], then∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ) ≤
∫
Ω2

F(f(s))ν(s)dµ2(s).

Proof.

• From right hand side of (32) we can write∫
Ω2

L1(f(s))ν(s)dµ2(s)−
∫
Ω1

L1(Akf(ϖ))u(ϖ)dµ1(ϖ).

Since L1 is convex, using Theorem 1 we find that the last relation is nonnegative.
Consequently, our stated inequality results from (32).

• Analogously, the assertion for the functional L2 can be derived from (33).

4. Bounds on Remainders

In this section, we make use of the Theorems 2 and 3 to discuss the Grüss and
Ostrowski-type inequalities and bounds on the remainders. Consider the following no-
tations for the sake of brevity.

B1(τ) =

∫ a2

a1

JGα(f, τ)× (τ − η)n−4
+ dτ (38)

and

B2(τ) = (−1)n−4

∫ a2

a1

JGα(f, τ)× (τ − η)n−4
+ dτ. (39)

Theorem 7. Let F be defined on T , for n ∈ N, F (n−1) has the property that it is ab-
solutely continuous therein and (τ − a1)(a2 − τ)[F (n+1)]2 ∈ L1([a1, a2]). Let (

∑
1,Ω1, σ1)

and (
∑

2,Ω2, σ2) with positive σ−finite measures, Ak and K are given in (15) and (16)
respectively. Let Gα, {α = 1, 2, 3, 4} be defined in (5), (6), (7) and (8) respectively, JGα

is defined in (26) u : Ω1 → R be the weight function and v is given in (17) and B1,B2 are
given in (38) and (39) respectively and f is measurable, then;

(i) the remainder R1 is

R1(F ; a1, a2)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 12 of 18

=

∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

−
∑
κ

F (κ)(a1)

κ!

∫ a2

a1

JGα(f, τ)(τ − a1)
κdτ

−F (n−1)(a2)−F (n−1)(a1)

(a2 − a1)(n− 4)!

∫ a2

a1

B1(τ)dτ, (40)

bounded by

|R1(F ; a1, a2)| ≤
√
a2 − a1√
2(n− 4)!

(
F(B1,B1)

) 1
2

(∫ a2

a1

(τ − a1)(a2 − τ)[F (n+1)]2dτ
) 1

2
.(41)

(ii) and the remainder R2 is

R2(F ; a1, a2) =

∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

−
∑
κ

(−1)κF (κ)(a2)

κ!

∫ a2

a1

JGα(f, τ)(a2 − τ)κdτ

−F (n−1)(a2)−F (n−1)(a1)

(a2 − a1)(n− 4)!

∫ a2

a1

B2(τ)dτ, (42)

bounded by

|R2(F ; a1, a2)| ≤
√
a2 − a1√
2(n− 4)!

(
F(B2,B2)

) 1
2

(∫ a2

a1

(τ − a1)(a2 − τ)[F (n+1)]2dτ
) 1

2
.(43)

Proof.

• From (24) and (40) we have

R1(F ; a1, a2) =
1

(n− 4)!

(∫ a2

a1

B1(τ)F (n)(τ)dτ − F (n−1)(a2)−F (n−1)(a1)

(a2 − a1)

∫ a2

a1

B1(τ)dτ

)
.(44)

Taking f = B1 and g = F (n) along with (44), then using Theorem 2 we obtain

1

a2 − a1

∣∣∣R1(F ; a1, a2)
∣∣∣

=
1

(n− 4)!

∣∣∣∣∣ 1

a2 − a1

∫ a2

a1

B1(τ)F (n)(τ)dτ − 1

(a2 − a1)

∫ a2

a1

B1(τ)dτ
1

(a2 − a1)

∫ a2

a1

F (n)(τ)dτ

∣∣∣∣∣
≤ 1√

2(n− 4)!

1√
a2 − a1

(
F(B1,B1)

) 1
2

(∫ a2

a1

(τ − a1)(a2 − τ)[F (n+1)]2dτ
) 1

2
. (45)

After simplification of involved integral we get the required result.



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 13 of 18

• Following similar steps of first part we obtain the required result.

Now we use Theorem 3 to discuss the Grüss-type inequalities that enable us to find
the bounds of remainder.

Theorem 8. Let (
∑

1,Ω1, σ1) and (
∑

2,Ω2, σ2) with positive σ−finite measures and Ak

and K be given in (15) and (16) respectively. Let F be defined on T such that for n ∈ N,
F (n) is absolutely continuous and F (n+1) ≥ 0 therein. Let Gα, {α = 1, 2, 3, 4} be defined
in (5), (6), (7) and (8) respectively and u : Ω1 → R be the weight function and v is given
in (17), B1,B2 are given in (38) and (39) respectively such that B′

1 and B′
2 ∈ L∞([a1, a2]),

then;

|R1(F ; a1, a2)| ≤
(a2 − a1)||B

′
1||∞

(n− 4)!

[
F (n−1)(a2) + F (n−1)(a1)

2
− F (n−2)(a2)−F (n−2)(a1)

a2 − a1

]
,(46)

|R2(F ; a1, a2)| ≤
(a2 − a1)||B

′
2||∞

(n− 4)!

[
F (n−1)(a2) + F (n−1)(a1)

2
− F (n−2)(a2)−F (n−2)(a1)

a2 − a1

]
.(47)

where R1(F ; a1, a2) and R2(F ; a1, a2) are given in (40) and (42) respectively.

Proof.

• As all the conditions stated in Theorem 3 are obayed if we take f = B1 and g = F (n).
So considering (44) we can have

1

a2 − a1

∣∣∣R1(F ; a1, a2)
∣∣∣ = 1

(n− 4)!

∣∣∣∣∣ 1

a2 − a1

∫ a2

a1

B1(τ)F (n)(τ)dτ

− 1

(a2 − a1)

∫ a2

a1

B1(τ)dτ
1

(a2 − a1)

∫ a2

a1

F (n)(τ)dτ

∣∣∣∣∣
≤ 1

2(n− 4)!

||B′
1||∞

a2 − a1

(∫ a2

a1

(τ − a1)(a2 − τ)F (n+1)(τ)dτ
)
. (48)

After simplifying integral in last step and taking (44) into account we get the required
result.

• Following smiliar steps of first part we obtain the required result.

Now moving forward for the analysis of Ostrowski-type inequalities related to the case
under consideration i.e generalized Hardy-type inequalities of convex functions via Taylor
polynomial and Green function



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 14 of 18

Theorem 9. Let all the conditions of Theorem 4 holds with n ∈ N and n ≥ 4. Let
JGα, B1 and B2 be given in (26), (38) and (39) respectively. Assuming that p, q be the
conjugate exponents with 1 ≤ p, q ≤ ∞ such that 1

q + 1
p = 1 and F : T → R be such that

||F (n)||p < ∞. Then

∣∣∣∣∣
∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

−
∑
κ

F (κ)(a1)

κ!

∫ a2

a1

JGα(f, τ)ν(s)dµ2(s)(τ − a1)
κ

∣∣∣∣∣ ≤ 1

(n− 4)!
||F (n)||p||B1||q (49)

and

∣∣∣∣∣
∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

−
∑
κ

F (κ)(a2)

κ!

∫ a2

a1

JGα(f, τ)(τ − a2)
κdτ

∣∣∣∣∣ ≤ 1

(n− 4)!
||F (n)||p||B1||q. (50)

Proof.

• Using Hölder inequality in (24) we arrive at∣∣∣∣∣
∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

−
∑
κ

F (κ)(a1)

κ!

∫ a2

a1

JGα(f, τ)(τ − a1)
κdτ

∣∣∣∣∣ = 1

(n− 4)!

∣∣∣ ∫ a2

a1

B1(τ)F (n)(τ)dτ
∣∣∣

≤ 1

(n− 4)!
||F (n)||p

(∫ a2

a1

|B1(τ)|qdτ
) 1

q
.

From this we can write (49).

• Following similar steps of part one after using Hölder inequality in (25) we get the
required result.

5. Mean Value Theorem (MVT) and n-Exponential Convexity

It is clear that inequalities (32) and (33) are linear in F . Keeping in mind all the
assumptions stated in Theorem 5, two linear functionals can be defined as following;

H1 =

∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 15 of 18

−
∫ a2

a1

JGα(f, τ)×
∑
κ

F (κ)(a1)

κ!
(τ − a1)

κdτ (51)

and

H2 =

∫
Ω2

F(f(s))ν(s)dµ2(s)−
∫
Ω1

F(Akf(ϖ))u(ϖ)dµ1(ϖ)

−
∫ a2

a1

JGα(f, τ)×
∑
κ

(−1)κ
F (κ)(a2)

κ!
(a2 − τ)κdτ. (52)

For any n-convex function F ∈ Cn([a1, a2]), we have Hγ(F) ≥ 0, for γ = 1, 2. Em-
ploying the linearity and non-negativity of the functionals given above, we can obtain the
corresponding MVT.

Theorem 10. Consider the aforementioned functionals Hγ, γ = 1, 2 defined in (51)and
(52) and F ∈ Cn([a1, a2]). Then there exist a1 ≤ cγ ≤ a2 such that

Hγ(F) = F (n)(cγ)Hγ(F0) γ = 1, 2, (53)

where F0(x) =
xn

n!

Proof. Denote m = minF (n) and M = maxF (n). Firstly, we consider H(ϖ) =
Mϖn

n! − F(ϖ). Then H(n)
1 (ϖ) = M − Fn ≥ 0, ϖ ∈ T . So we can say H1 is n-convex

function. Similarly, H2(ϖ) = F(ϖ) − mϖn

n! is n-convex. Following similar steps of [3,
Theorem 7] for convex functions H1 and H2 we have that there exist cγ for which (53)
holds.

Theorem 11. Let F1,F2 ∈ Cn([a1, a2]) and Hγ, γ = 1, 2 be defined in (51) and (52).
Then there exist cγ, γ = 1, 2 such that

F (n)
1 (cγ)

F (n)
2 (cγ)

=
Hγ(F1)

Hγ(F2)
, γ = 1, 2, (54)

for denominators that are not equal to zero.

Proof. The proof follows from [3, Corollary 12]

The seminal concept of obtaining n-exponentially convex and exponentially convex
functions was given in [11]. We apply aforementioned functionals on specific family given
in the upcoming theorem which is enough to get criteria for the exponential convexity of
the family of functions on selection of any distinct points.

Theorem 12. Consider Hγ , γ = 1, 2 be the functionals defined in (51) and (52). Let S =
{fµ : [a1, a2] → R} represents the class of functions having property that on choosing any of
r + 1 distinct points s0, ..., sr ∈ [a1, a2], the mapping µ → [s0, ..., sr; fµ] is n-exponentially
convex in Jensen sense on T so does the function µ → Hγ(fµ) on T . Furthermore, if
µ → Hγ(fµ) is continuous T , then it is n-exponentially convex on T .



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 16 of 18

Proof. Define a function for cj ∈ R and µj ∈ T , j = 1, ..., n and µk
j =

µi+µj

2 , 1 ≤ j, k ≤
n as;

ξ(x) =

n∑
j,k=1

cjckfµk
j
(x), (55)

where fµk
j
(x) ∈ S. Imploying the assumption that µ → [s0, ..., sn; fµ] is n-exponentially

convex in Jensen sense, we have

[s0, ..., sn; ξ] =
n∑

j,k=1

cjck[s0, ..., sn; fµk
j
] ≥ 0. (56)

Consequently, we have

Hγ(ξ) =
n∑

j,k=1

cjckHγ(fµk
j
) ≥ 0, (57)

for each γ = 1, 2. From this relation, it is evident that Hγ(fµ) is n-exponentially convex
in Jensen sense on T . Also if we take continuity of µ → Hγ(fµ) under consideration then
it is n-exponentially convex on T .

Some of the consequences of Theorem 12 can be obtained in the form of results given
in Corollary 4.4.1 and 4.4.2 in [15]

6. Conclusion

The study under consideration is the advancement and analysis of Hardy-type inequali-
ties using the two-point right focal problem-type Green functions and Taylor’s polynomial
that can be seen in Theorem 4, Theorem 5 and Theorem 6. Also, bounds on the re-
mainders are found in Theorem 7 by using Čebyšev functional. In the same section 4 we
discussed the Grüss-type inequalities in Theorem 8 to find the bound and Ostrowski-type
inequalities in Theorem 9 related to the Hardy-type inequalities via Taylor’s polynomial
and Green function. Next, we discuss the Mean value theorem for the functionals (51)
and (52) obtained from our main results and then make use of them to get results in the
form of MVT given in Theorem 10 and Theorem 11. Finally, using obtained functionals
(51) and (52) the n-exponential convexity is discussed as in Theorem 12.

Acknowledgements

The authors appreciate indebtedly, the anonymous referees for their meticulous reading
of the manuscript and for fruitful comments and suggestions.



A. M. K. Abbasi, M. Anwar / Eur. J. Pure Appl. Math, 18 (1) (2025), 5694 17 of 18

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