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Article

# An Application of Hayashi’s Inequality for Differentiable Functions

1
Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, P.O. Box 2600, Irbid 21110, Jordan
2
Faculty of Science, University of Split, Ruđera Boškovića 33, 21000 Split, Croatia
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(6), 907; https://doi.org/10.3390/math10060907
Submission received: 1 March 2022 / Revised: 9 March 2022 / Accepted: 10 March 2022 / Published: 11 March 2022

## Abstract

:
In this work, we offer new applications of Hayashi’s inequality for differentiable functions by proving new error estimates of the Ostrowski- and trapezoid-type quadrature rules.
MSC:
26D15

## 1. Introduction

In ([1], pp. 311–312) Hayashi proved the following theorem.
Theorem 1.
Let $p : [ a , b ] → R$ be a nonincreasing mapping on $[ a , b ]$ and $h : [ a , b ] → R$ an integrable mapping on $[ a , b ]$ with $0 ≤ h ( x ) ≤ A$ for all $x ∈ [ a , b ]$. Then, the inequality
$A ∫ b − λ b p ( x ) d x ≤ ∫ a b p ( x ) h ( x ) d x ≤ A ∫ a a + λ p ( x ) d x$
holds, where $λ = 1 A ∫ a b h ( x ) d x$.
Inequality (1), called Hayashi’s inequality, is a simple generalization of Steffensen’s inequality which holds with same assumptions with $A = 1$. For recent results concerning Hayashi’s inequality see [2].
In 1996, Agarwal and Dragomir [3] presented an application of this inequality as follows.
Theorem 2.
Let $f : I ⊆ R → R$ be a differentiable mapping on $I ∘$ (the interior of I) and $[ a , b ] ⊆ I ∘$ with , and $M > m$. If $f ′$ is integrable on $[ a , b ]$, then the following inequality holds
This elegant inequality presents an error estimation for the trapezoidal rule.
In 2002, Gauchman [4] generalized (2) for n-times differentiable functions using the Taylor expansion so that (2) becomes a special case of Gauchman’s result when $n = 0$.
In this paper, we present a generalization of (2). In the same argument, two other inequalities of the Ostrowski and trapezoidal type are also introduced.

## 2. The Results

Let us begin with a generalization of (2).
Theorem 3.
Let $g : [ a , b ] → R$ be an absolutely continuous function on $[ a , b ]$ with , and suppose that $g ′$ is integrable on $[ a , b ]$. Then
for all $x ∈ [ a , b ]$, where $λ = g b − g a b − a$. In particular, for $x = a + b 2$, the following inequality holds
Proof.
Let $f t = x − t$, $t ∈ [ a , b ]$. Applying Hayashi’s inequality (1) by setting $p t = f t$ and $h t = g ′ t$, we get
where, $A = b − a$ or we write
$λ = 1 b − a ∫ a b g ′ t d t = g b − g a b − a .$
Also, we have
and
Substituting the above equalities in (4) and dividing by $( b − a )$, we get
We also have
which proves the first inequality in (3). The second inequality follows by applying the same technique as in ([3], pp. 96–97). □
Remark 1.
For some results closely related to Theorem 3 we refer the reader to [5,6,7,8,9,10,11,12,13].
A corrected generalized version of the Agarwal-Dragomir result (2) is incorporated in the following corollary.
Corollary 1.
Let $g : [ a , b ] → R$ be an absolutely continuous function on $[ a , b ]$ with $γ ≤ g ′ t ≤ Γ$, and suppose that $g ′$ is integrable on $[ a , b ]$. Then
for all $x ∈ [ a , b ]$, where . In particular, for $x = a + b 2$, the following inequality holds
Proof.
Repeating the proof of Theorem 3, with $h ( t ) = g ′ ( t ) − γ$, $t ∈ [ a , b ]$, we get the first inequality. The second inequality in (5) follows by applying the same technique as in ([3], pp. 96–97). Analogous manipulation for $x = a + b 2$ gives the same result as in (2). □
Remark 2.
Let the assumptions of Corollary 1 be satisfied. Then Corollaries 3 and 4 and Remarks 1 and 2 in [3] (p. 97) also hold.
In 1997, Dragomir and Wang [11] introduced an inequality of Ostrowski-Grüss type as follows: inequality
holds for all $x ∈ [ a , b ]$, where f is assumed to be differentiable on $[ a , b ]$ with $f ′ ∈ L 1 [ a , b ]$ and $γ ≤ f ′ x ≤ Γ$, $∀ x ∈ [ a , b ]$.
In 1998, another result for twice differentiable was proved in [10]. In 2000, the constant $1 4$ in (6) was improved by $1 3$ in [14].
A better improvement of (6) can be deduced by applying Hayashi’s inequality as it is presented in the following theorem.
Theorem 4.
Let $g : [ a , b ] → R$ be an absolutely continuous function on $[ a , b ]$ with and suppose that $g ′$ is integrable on $[ a , b ]$. Then
for all $x ∈ [ a , b ]$, where $λ = g b − g a b − a$. In particular, for $x = a + b 2$, the following inequality holds
Proof.
Fix $x ∈ [ a , b ]$ and let $f t = a − t$, $t ∈ [ a , x ]$. Applying Hayashi’s inequality (1) by setting $p t = f t$ and $h t = g ′ t$, we get
where,
$λ = 1 b − a ∫ a b g ′ t d t = g b − g a b − a .$
Also, we have
and
$∫ a a + λ ( a − t ) d t = − 1 2 λ 2 .$
Substituting in (8), we get
Now, let $f t = b − t$, $t ∈ [ x , b ]$. Applying Hayashi’s inequality (1) again we get
where,
$∫ b − λ b ( b − t ) d t = 1 2 λ 2 ,$
and
Substituting in (10), we obtain
Adding (9) and (11) we get
Setting
$I : = 1 b − a ∫ a b g t d t − g x$
and
Therefore,
which proves the first inequality in (7). To prove the second inequality, define the mapping $ϕ ( t ) = − t 2 + b − a 2 t$. Then , so that , which completes the proof of this theorem. □
Corollary 2.
Let $g : [ a , b ] → R$ be an absolutely continuous function on $[ a , b ]$ with $γ ≤ g ′ t ≤ Γ$ and suppose that $g ′$ is integrable on $[ a , b ]$. Then
for all $x ∈ [ a , b ]$, where . In particular, for $x = a + b 2$, the following inequality holds
Proof.
Repeating the proof of Theorem 4, with $h ( t ) = g ′ ( t ) − γ$, $t ∈ [ a , b ]$, we get the first inequality. The second inequality (12) follows by applying the same technique. □
Remark 3.
As we notice, (12) improves (6) by $1 4$, which is better than Matić et al. result from [14].
In [6], under the assumptions of Theorem 4, Alomari proved the following version of a Guessab–Schmeisser-type inequality (see [12]):
for all .
Next we give an improvement of (13).
Theorem 5.
Let $g : [ a , b ] → R$ be an absolutely continuous function on $[ a , b ]$ with and suppose that $g ′$ is integrable on $[ a , b ]$. Then
for all , where $λ = g b − g a b − a$.
Proof.
Fix and let $f t = a − t$, $t ∈ [ a , x ]$. Applying Hayashi’s inequality (1) by setting $p t = f t$ and $h t = g ′ t$, we get that (8) holds, i.e.,
where
$λ = 1 b − a ∫ a b g ′ t d t = g b − g a b − a .$
Now, let $f t = a + b 2 − t$, . Applying Hayashi’s inequality (1) again we get
where
and
Substituting in (16), we get
Now, let $f t = b − t$, $t ∈ [ a + b − x , b ]$. Applying Hayashi’s inequality (1) again we obtain
where
$∫ b − λ b ( b − t ) d t = 1 2 λ 2 ,$
and
Substituting in (18) we get
Adding (15), (17) and (19) we obtain
which is equivalent to the first inequality in (14). To prove the second inequality in (14), define the mapping . Then , so that , which completes the proof. □
A generalization of (13) and (14) is incorporated in the following result.
Corollary 3.
Let $g : [ a , b ] → R$ be an absolutely continuous function on $[ a , b ]$ with $γ ≤ g ′ t ≤ Γ$ and suppose that $g ′$ is integrable on $[ a , b ]$. Then
for all , where .
Proof.
Repeating the proof of Theorem 5, with $h ( t ) = g ′ ( t ) − γ$, $t ∈ [ a , b ]$, we get the first inequality. The second inequality in (20) follows by applying the same technique. □

## 3. Applications

Let X be a random variable taking values in the finite interval $[ a , b ]$, with the probability density function $f : [ a , b ] → [ 0 , 1 ]$ with the cumulative distribution function $F ( x ) = P r ( X ≤ x ) = ∫ a b f ( t ) d t$.
Theorem 6.
With the assumptions of Theorem 4, we have the inequality
for all , where , and $E ( X )$ is the expectation of X.
Proof.
In the proof of Corollary 3, we set $f = F$, and take into account that
$E X = ∫ a b t d F t = b − ∫ a b F t d t .$
We leave the details to the interested reader. □

## 4. Conclusions

This paper summarises several types of general quadrature rules, such as the general trapezoidal rule or the so-called Ostrowski trapezoidal, Ostrowski midpoint and Guessab–Schmeisser quadrature rules for symmetric points. Using the presented inequalities, several error estimates of the above quadrature rules could therefore be derived with corresponding numerical experiments. This work thus represents a very good application of Hayashi’s inequality in quadrature approximation. One future research direction might be to use Fink’s generalization of the Ostrowski inequality to obtain some Hayashi–Ostrowski-type results. Further applications of Hayashi’s inequality we leave to the interested reader.

## Author Contributions

Conceptualization, M.W.A.; Formal analysis, M.W.A. and M.K.B.; Funding acquisition, M.K.B.; Investigation, M.K.B.; Methodology, M.W.A.;Writing original draft preparation, M.W.A. and M.K.B. All authors have read and agreed to the published version of the manuscript.

## Funding

This publication was supported by the University of Split, Faculty of Science.

Not applicable.

Not applicable.

Not applicable.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

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Alomari, M.W.; Klaričić Bakula, M. An Application of Hayashi’s Inequality for Differentiable Functions. Mathematics 2022, 10, 907. https://doi.org/10.3390/math10060907

AMA Style

Alomari MW, Klaričić Bakula M. An Application of Hayashi’s Inequality for Differentiable Functions. Mathematics. 2022; 10(6):907. https://doi.org/10.3390/math10060907

Chicago/Turabian Style

Alomari, Mohammad W., and Milica Klaričić Bakula. 2022. "An Application of Hayashi’s Inequality for Differentiable Functions" Mathematics 10, no. 6: 907. https://doi.org/10.3390/math10060907

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