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On some inequalities relative to the Pompeiu–Chebyshev functional
Journal of Inequalities and Applications volume 2020, Article number: 46 (2020)
Abstract
In this paper we study the utility of the functional Pompeiu–Chebyshev in some inequalities. Some results obtained by Alomari will be generalized regarding inequalities with Pompeiu–Chebyshev type functionals, in which linear and positive functionals intervene. We investigate some new inequalities of Grüss type using Pompeiu’s mean value theorem. Improvement of known inequalities is also given.
1 Introduction
If f, g are integrable functions on \([a,b] \), then the functional defined by
is known as functional Chebyshev, with multiple applications in numerical analysis and probability theory (see [3]).
The following theorem combines a series of results regarding the bounds for this functional.
Theorem 1.1
Let\(f,g:[a,b]\rightarrow \mathbb{R} \)be two absolutely continuous functions, then
In [9], Pompeiu established the following mean value theorem for functions defined on an interval \([a,b] \) such that \(0\notin [a,b] \).
Theorem 1.2
For every function\(f\in C^{1}[a,b]\), \(0\notin [a,b]\)and for all\(x,y\in [a,b]\), \(x\neq y \), there is\(c\in (x,y) \)such that
From (3), we obtain that
where \(e_{i}=x^{i}\), \(i=\overline{0,n}\), \(n\in \mathbb{N} \).
In 2005, Pachpatte (see [8]) introduced the following functional.
If \(f,g:[a,b]\rightarrow \mathbb{R} \) are two differentiable functions on \((a,b) \), then
and proved the following result.
Theorem 1.3
If\(f,g:[a,b]\rightarrow \mathbb{R} \)are two continuous on\([a,b] \)and differentiable functions on\((a,b) \)such that\(0\notin [a,b] \), then
Dragomir (see [4]) studied the Pompeiu–Chebyshev functional and changed it as follows:
The following result, obtained by Dragomir in [4], will be used in some demonstrations included in this paper.
Lemma 1.4
Let\(f:[a,b]\rightarrow \mathbb{R} \)be an absolutely continuous function on\([a,b]\), \(b>a>0 \). Then, for any\(x,y\in [a,b] \), we have
where\(p,q>1 \)with\(\frac{1}{p}+ \frac{1}{q}=1 \).
In [1], Alomari studied and generalized some inequalities related to the Pompeiu–Chebyshev functional.
The purpose of this paper is to generalize the results of Alomari considering the Pompeiu–Chebyshev functional in which linear and positive functionals intervene.
2 Main results
In the following we denote by \(\mathcal{F} \) a set of linear functions defined on the interval \(I=[a,b] \). We will assume that the set \(\mathcal{F} \) contains the constant and polynomial functions, and we suppose that, if \(f,g\in \mathcal{F}\), then \(f\cdot g\in \mathcal{F} \).
Definition 2.1
Let \(A,B:\mathcal{F}\rightarrow \mathbb{R} \) be two linear and positive functionals.
If \(f,g\in \mathcal{F}\), we denote
We call the functional \(\mathcal{P}_{A,B}(f,g) \) a Pompeiu–Chebyshev functional.
Remark 2.2
For any two linear and positive functionals \(A,B:\mathcal{F}\rightarrow \mathbb{R} \), we have
Remark 2.3
If we take
then the functional \(\mathcal{P}_{A,B}(f,g) \) becomes the functional that was studied by Dragomir in [4].
Theorem 2.4
If\(\mathcal{F}=C^{1}[a,b]\), \(0\notin [a,b] \), then
Proof
From Lemma 1.4 we have
Next, by \(A_{x} \) or \(B_{y} \) we will understand that the functional A, respectively B, acts on the variable x, respectively y.
It is easy to see that
From relations (9) and (10) we get the following:
Further we have
Combining relations (11) and (12), we get (8). □
Corollary 2.5
If we take\(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we obtain
so,
which is the inequality obtained by Dragomir in [4].
Definition 2.6
Let \(f,g\in \mathcal{ F} \). The functions f and g are called synchronous (or similarly ordered) if for all \(x,y\in I \), where I is the domain for f and g, we have
and f and g are called asynchronous (or oppositely ordered) if for all \(x,y\in I \) we have
Theorem 2.7
Let\(f,g\in \mathcal{ F} \), where\(f,g:I\rightarrow \mathbb{R} \)such that\(0\notin I \).
- (i)
If\(\frac{f}{e_{1}} \)and\(\frac{g}{e_{1}} \)are synchronous functions, then\(\mathcal{P}_{A,B}(f,g) \geq 0\).
- (ii)
If\(\frac{f}{e_{1}} \)and\(\frac{g}{e_{1}} \)are asynchronous functions, then\(\mathcal{P}_{A,B}(f,g) \leq 0 \).
Proof
Since \(\frac{f}{e_{1}} \) and \(\frac{g}{e_{1}} \) are synchronous (asynchronous) functions, we have
So,
□
Remark 2.8
If \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we get Theorem 6 and Corollary 1 from [1].
The following theorem shows a pre-Grüss inequality for the functional \(\mathcal{P}_{A,B}(f,g) \) (see [5]).
Theorem 2.9
Let\(f,g\in \mathcal{ F} \), where\(f,g:I\rightarrow \mathbb{R} \). Then
Constant 1 is the best possible.
Proof
From the equality
and from the CBS-inequality, we obtain
But we have
and
From the above the conclusion is obtained. □
We notice that for \(f(x)=g(x)=c\cdot x-1\), \(c\in \mathbb{R} \), fixed we obtain the equality in (16).
We note that for \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \) we get Theorem 7 from [1].
Theorem 2.10
Let\(f,g:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F} \). If there exist real numbers\(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g} \)such that\(m_{f}\leq f(x)\leq M _{f} \)and\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in [a,b] \), then the following inequality
holds.
Proof
From the assumptions of the theorem we have
Adding the last inequalities, we have
or
In the same way we proceed for the function g, and we get
From (18) we get
So, we have
Since
we get the inequality from the conclusion. □
Corollary 2.11
If we take\(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we get the following inequality:
Theorem 2.12
Let\(A,B:\mathcal{F}\rightarrow \mathbb{R} \)be two linear and positive functionals. Let\(f,g:[a,b] \rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F} \). If there exist real numbers\(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g} \)such that\(m_{f}\leq f(x)\leq M_{f} \)and\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in [a,b] \), then the following inequality holds:
Proof
From the assumptions of the theorem we have
Adding the last inequalities, we have
or
In the same way we proceed for the function g, and we get
From the above we get
and then
The last inequality is equivalent to the conclusion. □
Remark 2.13
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we get Theorem 8 from [1].
Theorem 2.14
Let\(f,g:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F} \). If there exist real numbers\(m_{g}\), \(M_{g} \)such that\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x \in [a,b] \), then the following inequality
holds.
Proof
From (16) we get
From relationships (22) and (23) we get the conclusion. □
Corollary 2.15
If we take\(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we obtain the following inequality:
Theorem 2.16
Let\(A,B:\mathcal{F}\rightarrow \mathbb{R} \)be two linear and positive functionals. Let\(f,g:[a,b] \rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F} \). If there exist real numbers\(m_{g}\), \(M_{g} \)such that\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in [a,b] \), then the following inequality holds:
Proof
From (16) we get
From (17) for \(f=g \) we have
Replacing (27) in (26), we get the conclusion. □
Remark 2.17
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we get Theorem 9 from [1].
An improvement of inequality (17) from Theorem 2.10 is given below.
Theorem 2.18
LetDbe a subset of the real line such that\(D\subset [a,b]\), \(a>0 \). If\(f,g\in \mathcal{F}\), \(f,g:D\rightarrow \mathbb{R}\), \(0< a< b \)and we suppose that there exist real numbers\(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g} \)such that\(m_{f}\leq f(x)\leq M_{f} \)and\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in D \), then the following inequality holds:
Proof
It is easy to see, in these conditions, that we have
From the above we obtain
Applying the linear and positive functional \(A_{x}B_{y} \) and considering that
the statement results. □
Remark 2.19
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then from (22) we get Theorem 10 from [1].
A generalization of this is given in what follows.
Theorem 2.20
Let\(f,g:D\rightarrow \mathbb{R}\), \(D\subset [a,b]\), \(0< a< b\), \(f,g\in \mathcal{F} \). If there exist real numbers\(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g}\)such that\(m_{f}\leq f(x)\leq M _{f} \)and\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in D \), then the following inequality holds:
where
and
Proof
Using the Cauchy–Schwarz inequality in (28), we have
From the above we obtain
Computing, we obtain
and
So, we get
where \(K_{f} \) and \(K_{g} \) are given in (30), respectively (31), which is the inequality from the conclusion. □
A more general case is taken forward, which improves relationship (25).
Theorem 2.21
Let\(f,g:D\rightarrow \mathbb{R}\), \(D \subset [a,b]\), \(0< a< b\), \(f,g\in \mathcal{F} \). If there exist real numbers\(m_{g}\), \(M_{g}\)such that\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in D \), then
where\(K_{g} \)is given by (31).
Proof
Using inequality (29), we have
and replacing this in relation (16), we get inequality (24). □
Remark 2.22
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then we obtain
which is inequality (2.14) from [1, Th. 11].
3 Applications
In this section we investigate some new inequalities of Grüss type using Pompeiu’s mean value theorem and the above results. Improvement of known inequalities is also given.
Theorem 3.1
Let\(f,g:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F} \). If\(f\in C^{1}[a,b] \), then the following inequality
holds.
Proof
From Lemma 1.4 we have that
and it follows that
□
Remark 3.2
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then from (29) we get the first result from [1, Th. 13].
Theorem 3.3
Let\(f,g:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F} \). If\(f\in L^{1}[a,b] \), then the following inequality
holds.
Proof
From Lemma 1.4 we have
□
Remark 3.4
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then from (36) we get the last result from [1, Th. 13].
Theorem 3.5
Let\(f,g:D \rightarrow \mathbb{R}\), \(D \subset [a,b]\), \(0< a< b\), \(f,g\in \mathcal{F}\), \(f\in C^{1}[a,b] \). If there exist real numbers\(m_{g}\), \(M_{g}\)such that\(m_{g}\leq g(x)\leq M _{g}\), \(\forall x\in [a,b] \), then
where\(K_{g} \)is given by (31).
Proof
From inequality (8) we get
On the other hand, from inequality (29) we have
Using the last two inequalities in (16), we get the conclusion. □
Remark 3.6
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then from (37) we get the result from [1, Th. 14].
Theorem 3.7
Let\(f:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F}\), \(f\in C^{1}[a,b]\), \(g\in L^{1}[a,b]\). Then we have the following inequality:
Proof
From Lemma 1.4 we get
If \(x< y \), then
If \(x\geq y \), then
Therefore, we have
Using the last inequality in (39), we obtain (38). □
Remark 3.8
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (39) becomes the following inequality:
Theorem 3.9
Let\(f:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F}\), \(f\in C^{1}[a,b]\), \(g\in L^{1}[a,b] \). Then we have the following inequality:
Proof
From Lemma 1.4 we get
□
Remark 3.10
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (41) becomes inequality (3.5) from [1, Th. 15].
Theorem 3.11
Let\(f:[a,b]\rightarrow \mathbb{R}\), \(0< a< b\), \(f,g\in \mathcal{F}\), \(f\in L^{1}[a,b]\). If there exist real numbers\(m_{g}\), \(M_{g}\)such that\(m_{g}\leq g(x)\leq M_{g}\), \(\forall x \in [a,b] \), then the following inequality holds:
where\(K_{g} \)is given by (31).
Proof
From Lemma 1.4 we get
Using inequality (29), we have
where \(K_{g} \) is given by (31).
Substituting in (16) we get the desired result. □
Remark 3.12
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (42) becomes inequality (3.7) from [1, Th. 16].
Definition 3.13
Let \(a,b\in \mathbb{R} \), with \(a< b \) and \(f,g,h\in \mathcal{ F}\), \(h:[a,b]\rightarrow \mathbb{R}_{+} \). The functional noted by \(\mathcal{P}_{A,B}(f,g;h) \), defined by
is called Pompeiu–Chebyshev with respect to the function h functional.
Note that the functional can also be written in the following form:
Definition 3.14
(See [1])
Let \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g \in \mathcal{ F} \). The functions f and g are called synchronous with respect to a function h (h-synchronous, similarly ordered), \(h:[a,b]\rightarrow \mathbb{R}_{+} \), if for all \(x,y\in [a,b] \), we have
and f, g are called asynchronous with respect to a function h (h-asynchronous, oppositely ordered) if for all \(x,y\in [a,b] \) we have
The next result generalizes the inequalities from Theorem 2.7.
Theorem 3.15
Let\(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{ F} \), and\(h:[a,b]\rightarrow \mathbb{R}_{+} \)such that\(h(x)\neq 0\), \(\forall x\in [a,b] \).
- (i)
If\(\frac{f}{h} \)and\(\frac{g}{h} \)areh-synchronous functions, then
$$ \mathcal{P}_{A,B}(f,g;h)\geq 0. $$(46) - (ii)
If\(\frac{f}{h} \)and\(\frac{g}{h} \)areh-asynchronous functions, then
$$ \mathcal{P}_{A,B}(f,g;h)\leq 0. $$(47)
Proof
Since \(\frac{f}{h} \) and \(\frac{g}{h} \) are h-synchronous (h-asynchronous) functions, we have
From this and (43) we have
from where we get the conclusion. □
Remark 3.16
In (46), respectively (47), if we take \(h(x)=x\), \(x\in [a,b] \), then we obtain the inequalities from Theorem 2.7.
Remark 3.17
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (46) becomes inequality (4.5) from [1, Th. 19].
The next theorem is a generalization of Theorem 2.9 and contains the pre-Grüss inequality.
Theorem 3.18
Let\(f,g:[a,b]\rightarrow \mathbb{R}\), \(a,b\in \mathbb{R}\), \(a< b \), and\(f,g\in \mathcal{F} \). If\(h\in \mathcal{F} \)is a positive function, then
Proof
Using the CBS inequality in equality (43), we obtain
But we have
and
From the above we get the conclusion. □
Remark 3.19
In (46), if we take \(h(x)=x\), \(x\in [a,b] \), then we obtain inequality (16) from Theorem 2.9.
Remark 3.20
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (46), respectively (47), becomes inequalities (4.8) from [1, Corollary 8].
It is easy to see that the Pompeiu–Chebyshev functional with respect to the function h, \(\mathcal{P}_{A,B}(f,f;h) \) represents the reverse of CBS-inequality. We have
We recall that a function \(f:[a,b]\rightarrow \mathbb{R} \) is called of \(p-H\)-Hölder type, with \(H>0\), \(p\in (0,1] \), if for any \(x,y\in [a,b] \) we have
In [2], Barnett and Dragomir proved the following theorem.
Theorem 3.21
Iff, gare measurable on\([a,b] \)and\(\frac{f}{g} \)is\(p-H\)-Hölder type, with\(H>0\), \(p\in (0,1]) \), then
for\(\alpha ,\beta >1\), \(\frac{1}{\alpha }+ \frac{1}{\beta }=1 \)and\(\Vert h \Vert _{p}= ( \int _{a}^{b} \vert f(x) \vert ^{p}\,dx )^{\frac{1}{p}} \).
Starting from this we can state the following results.
Theorem 3.22
Let\(a,b\in \mathbb{R}\), \(a< b \), and\(f,g,h\in \mathcal{F} \), where\(f,g,h:[a,b]\rightarrow \mathbb{R} \). If\(\frac{f}{h} \)and\(\frac{g}{h} \)are of\(p-H\)-Hölder type, with\(H_{1},H_{2}>0\), \(p,q\in (0,1] \), then
Proof
From \(\frac{f}{h} \) and \(\frac{g}{h} \) are of \(p-H\)-Hölder type with \(H_{1},H_{2}>0\), \(p,q \in (0,1] \) we have
and
By multiplying the last two inequalities, we obtain
Using (43) in the last inequality \(\mathcal{P}_{A,B}(f,g;h) \), we obtain the conclusion. □
Remark 3.23
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (41) becomes inequality (4.13) from [1, Theorem 20], which represents the following inequalities:
Theorem 3.24
Let\(a,b\in \mathbb{R}\), \(a< b \), and\(f,g,h\in \mathcal{F} \), where\(f,g,h:[a,b]\rightarrow \mathbb{R} \). If\(\frac{h}{f} \)and\(\frac{h}{g} \)are of\(p-H\)-Hölder type, with\(H_{1},H_{2}>0\), \(p,q\in (0,1] \), then
Proof
From \(\frac{h}{f} \) and \(\frac{h}{g} \) are of \(p-H\)-Hölder type with \(H_{1},H_{2}>0\), \(p,q \in (0,1] \) we have
and
By multiplying the last two inequalities, we obtain
Applying in the last inequality \(\mathcal{P}_{A,B}(f,g;h) \), we obtain the conclusion. □
Remark 3.25
If we take \(A(f)=B(f)= \int _{a}^{b}f(x)\,dx \), then inequality (43) becomes (4.16) from [1, Theorem 21], which represents the following inequalities:
4 Examples
In this section we give some examples by choosing the functionals \(A(f) \) and \(B(f) \) in different forms and, in this way, we obtain some inequalities.
Example 4.1
Let
be two functionals for which we have
and
Using (7) for the functionals A and B chosen, we obtain
For the functional defined by (56), we obtain the following inequalities:
- (a)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F}\), \(f,g\in C^{1}[a,b] \), then
$$ \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq \frac{(b-a)^{2}}{6} \bigl\Vert f-e_{1}f' \bigr\Vert _{\infty } \bigl\Vert g-e_{1}g' \bigr\Vert _{ \infty }. $$(57) - (b)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F} \), and there exist real numbers \(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g}\) such that \(m_{f}\leq f(x)\leq M_{f} \) and \(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in D \), then
$$ \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq \frac{(b-a)^{2}}{6} (bM _{f}-am_{f}) (bM_{g}-am_{g}). $$(58)
Example 4.2
Let
be a Riemann–Liouville type functional for which we have
For \(\alpha =1 \) we denote
and for \(\alpha =2 \) we denote
We have
and
Substituting (60) and (61) in (7), we obtain
For the functional defined by (62), we obtain the following inequalities:
- (a)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F}\), \(f,g\in C^{1}[a,b] \), then
$$ \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq \frac{(b-a)^{5}}{24} \bigl\Vert f-e_{1}f' \bigr\Vert _{\infty } \bigl\Vert g-e_{1}g' \bigr\Vert _{\infty }. $$(63) - (b)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F} \), and there exist real numbers \(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g} \) such that \(m_{f}\leq f(x)\leq M_{f} \) and \(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in [a,b] \), then
$$ \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq \frac{(b-a)^{5}}{24} (bM _{f}-am_{f}) (bM_{g}-am_{g}). $$(64) - (c)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F} \), and there exist real numbers \(m_{g}\), \(M_{g} \) such that \(m_{g} \leq g(x)\leq M_{g}\), \(\forall x\in [a,b] \), then
$$ \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq \frac{(b-a)^{\frac{5}{2}}}{2 \sqrt{6}} \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert ^{\frac{1}{2}} (bM _{g}-am_{g}). $$(65)
Example 4.3
Let
and
be two Riemann–Liouville type functionals for which we have
where \(\varphi \in \{ \alpha ,\beta \} \).
For \(R_{\alpha }(f)=A(f) \) and \(R_{\beta }(f)=B(f) \), using relations (66) in (7), we have
For the functional defined by (67), we obtain the following inequalities:
- (a)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F}\), \(f,g\in C^{1}[a,b] \), then
$$ \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq \frac{(\alpha ^{2} + \beta ^{2} + \alpha + \beta - \alpha \beta )(b - a)^{\alpha +\beta +2}}{(\alpha +2)!(\beta +2)!} \bigl\Vert f-e_{1}f' \bigr\Vert _{\infty } \bigl\Vert g-e_{1}g' \bigr\Vert _{\infty }. $$(68) - (b)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F} \), and there exist real numbers \(m_{f}\), \(M_{f}\), \(m_{g}\), \(M_{g} \) such that \(m_{f}\leq f(x)\leq M_{f} \) and \(m_{g}\leq g(x)\leq M_{g}\), \(\forall x\in [a,b] \), then
$$\begin{aligned} \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq{}& \frac{(\alpha ^{2}+\beta ^{2}+\alpha +\beta -\alpha \beta )(b-a) ^{\alpha +\beta +2}}{(\alpha +2)!(\beta +2)!} \\ &{} \times (bM_{f}-am_{f}) (bM_{g}-am_{g}). \end{aligned}$$(69) - (c)
If \(0< a< b\), \(f,g:[a,b]\rightarrow \mathbb{R}\), \(f,g\in \mathcal{F} \), and there exist real numbers \(m_{g}\), \(M_{g}\) such that \(m_{g}\leq g(x) \leq M_{g}\), \(\forall x\in [a,b] \), then
$$\begin{aligned} \bigl\vert \mathcal{P}_{A,B}(f,g) \bigr\vert \leq{}& \biggl[\frac{(\alpha ^{2}+\beta ^{2}+\alpha +\beta -\alpha \beta )(b-a)^{\alpha +\beta +2}}{( \alpha +2)!(\beta +2)!} \biggr]^{\frac{1}{2}} \\ &{} \times \bigl\vert \mathcal{P}_{A,B}(f,f) \bigr\vert ^{\frac{1}{2}} (bM _{g}-am_{g}). \end{aligned}$$(70)
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The authors are grateful to the PhD coordinator, Prof. Ioan Gavrea, Department of Mathematics, Technical University of Cluj-Napoca, Romania for his careful reading of the manuscript and recommendations which improved the quality of the paper.
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Ianoşi, D., Opriş, AA. On some inequalities relative to the Pompeiu–Chebyshev functional. J Inequal Appl 2020, 46 (2020). https://doi.org/10.1186/s13660-020-02312-0
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DOI: https://doi.org/10.1186/s13660-020-02312-0