An extension of Schweitzer's inequality to Riemann-Liouville fractional integral

Thabet Abdeljawad; Badreddine Meftah; Abdelghani Lakhdari; Manar A. Alqudah

doi:10.1515/math-2024-0043

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An extension of Schweitzer's inequality to Riemann-Liouville fractional integral

Thabet Abdeljawad , Badreddine Meftah , Abdelghani Lakhdari and Manar A. Alqudah

Published/Copyright: August 26, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

Keywords: Schweitzer inequality; concave functions; Cauchy-Schwarz inequality; Bernoulli inequality; Riemann-Liouville integral operator

MSC 2010: 26D10; 26D15; 26A51

1 Introduction

Integral inequalities are a fundamental aspect of mathematical analysis, particularly in the realm of calculus and mathematical modeling. These inequalities establish relationships and bounds between integrals of functions, offering essential tools to quantify and understand the properties of various mathematical expressions. From classical integral inequalities like the Cauchy-Schwarz inequality to more advanced inequalities such as Hölder’s inequality and Minkowski’s inequality, these tools are pivotal in fields ranging from physics to economics. They play a significant role in optimizing problem-solving techniques, providing a pathway to efficiently analyze and solve complex mathematical problems.

In 1914, Schweitzer [1] established the following inequality:

For integrable functions $ℋ$ and $1 ℋ$ such that $0 < m ≤ ℋ ( x ) ≤ M$ on $[ a , b ]$ , we have

(1)

$1 b − a ∫ a b ℋ ( u ) d u 1 b − a ∫ a b 1 ℋ ( u ) d u ≤ ( m + M ) 2 4 m M .$

This inequality, known as the Schweitzer inequality, stands as a fundamental result in mathematical analysis, illuminating a critical link between integrals and functions. Its far-reaching importance and diverse applications establish it as a foundational tool across various mathematical and scientific domains.

In the same article, Schweitzer also presented a discrete version of (1) as follows:

(2)

$1 n ∑ k = 1 n a k 1 n ∑ k = 1 n 1 a k ≤ ( m + M ) 2 4 m M ,$

where $0 < m ≤ a k ≤ M , k = 1 , … , n$ .

In 1948, Kantorovič [2] provided the estimation of inequality (2) for both the left and right sides, as follows:

(3)

$1 ≤ 1 ∑ k = 1 n μ k ∑ k = 1 n μ k a k 1 ∑ k = 1 n μ k ∑ k = 1 n μ k a k ≤ ( m + M ) 2 4 m M .$

Wilkins [3] demonstrated that for a concave, monotone function $ℋ$ on $( 0 , 1 )$ satisfying $0 < m ≤ ℋ ( x ) ≤ M$ , we have

(4)

$1 ≤ ∫ 0 1 ℋ ( u ) d u ∫ 0 1 1 ℋ ( u ) d u ≤ M ( ln M − ln m ) M − m − m + M 2 M 2 2 ( M − m ) ln M − ln m M − m − 1 M .$

Brenner and Alzer [4] showed that if a positive function $ℋ$ is continuous and concave over the interval $[ a , b ]$ , then we have

(5)

$1 ≤ 1 ( b − a ) 2 ∫ a b ℋ ( u ) d u ∫ a b 1 ℋ ( u ) d u ≤ 1 + ln ℋ a + b 2 ℋ ( b ) ℋ ( a ) .$

Fractional calculus, a branch of mathematical analysis, extends traditional calculus to noninteger orders. It involves the study of derivatives and integrals of noninteger or fractional orders, providing a powerful framework to model various real-world phenomena with greater precision. The concept dates back to the eighteenth century, but has gained significant traction in recent decades due to its applicability in physics, engineering, biology and other disciplines [5–8]. Fractional calculus has proven instrumental in describing complex systems characterized by anomalous behaviors, offering insights that classical calculus may overlook.

Conversely, the Schweitzer inequality is one of the fundamental inequalities in classical calculus, and extending its fractional counterpart is a valuable pursuit. This note introduces a fractional analog of the Schweitzer inequality, specifically tailored for the left-sided Riemann-Liouville fractional integral operator.

2 Main results

We first recall the left-sided Riemann-Liouville fractional integral as well as some generalizations of Bernoulli’s inequality.

Definition 2.1

[9] Let $ℋ ∈ L 1 [ a , b ]$ with $a ≥ 0$ and $α > 0$ . The left-sided Riemann-Liouville fractional integral is as follows:

$I a + α ℋ ( b ) = 1 Γ ( α ) ∫ a b ( b − u ) α − 1 ℋ ( u ) d u , b > a ,$

where $Γ ( α ) = ∫ 0 ∞ e − t t α − 1 d t$ is the gamma function and $I a + 0 ℋ ( x ) = ℋ ( x )$ .

Lemma 2.2

[10] For $x > − 1$ , the following generalizations of Bernoulli’s inequality are valid:

$( 1 + x ) σ ≥ 1 + σ x for σ > 1 and σ < 0 , ( 1 + x ) σ ≤ 1 + σ x for 0 < σ < 1 .$

Theorem 2.3

Let $ℋ : [ a , b ] → R + = [ 0 , + ∞ )$ be continuously differentiable increasing and concave function, then for $α > 0$ , we have

(6)

$1 ≤ ( Γ ( α + 1 ) ) 2 ( b − a ) 2 α J a + α ( ℋ ) ( b ) J a + α ( h ∘ ℋ ) ( b ) ≤ ϱ ( α , a , b ) if 0 < α ≤ 1 , ς ( α , a , b ) if 1 < α ≤ 2 , υ ( α , a , b ) if α > 2 ,$

where

$ϱ ( α , a , b ) = 1 + α ( 1 − α ) b α ( b − a ) α − ( 2 − α ) α 2 b α − 1 ( α + 1 ) ( b − a ) α − 1 ln ℋ ( b ) + α 2 b α − 2 ( b − ( α − 1 ) a ) ( α + 1 ) ( b − a ) α − 1 − α b α − 1 ( b − α a ) ( b − a ) α ln ℋ ( a ) + b α − 1 ( 2 − α ) α ( b − a ) α + ( 1 − α ) α 2 b ( α + 1 ) ( b − a ) α − 1 ( b − a ) ln ℋ a + b 2 , ς ( α , a , b ) = 1 + ( α + 1 ) α b α − 1 − α 2 b α − 2 ( b − a ) ( α + 1 ) ( b − a ) α − 1 ln ℋ ( a ) ℋ a + b 2 , υ ( α , a , b ) = 1 + α 2 b α − 1 ( ( 3 − α ) b + ( 1 − α ) a ) 2 ( α + 1 ) b ( b − a ) α − 1 ln ℋ ( a ) ℋ ( b ) + α b α − 1 ( b − a ) α − 1 ln ℋ a + b 2 ℋ ( a )$

and $h ( t ) = 1 t$ .

Proof

To prove the first inequality of (6), we have from differentiability and positivity of $ℋ$

(7)

$( Γ ( α + 1 ) ) 2 ( b − a ) 2 α J a + α ( ℋ ) ( b ) J a + α ( h ∘ ℋ ) ( b ) = ( Γ ( α + 1 ) ) 2 ( b − a ) 2 α 1 Γ ( α ) ∫ b a ( b − x ) α − 1 ℋ ( x ) d x 1 Γ ( α ) ∫ b a ( b − x ) α − 1 1 ℋ ( x ) d x = α 2 ( b − a ) 2 α ∫ b a ( b − x ) α − 1 2 ℋ ( x ) 2 d x ∫ b a ( b − x ) α − 1 2 1 ℋ ( x ) 2 d x .$

By applying Cauchy-Schwarz inequality to (7), we obtain

$α 2 ( b − a ) 2 α ∫ b a ( b − x ) α − 1 2 ℋ ( x ) 2 d x ∫ b a ( b − x ) α − 1 2 1 ℋ ( x ) 2 d x ≥ α 2 ( b − a ) 2 α ∫ b a ( b − x ) α − 1 d x 2 = α 2 ( b − a ) 2 α ( b − a ) α α 2 = 1 .$

For the second inequality of (6). From differentiability of $ℋ$ , we have for all $x , y ∈ [ a , b ]$

(8)

$ℋ ( x ) = ℋ ( y ) + ( x − y ) ℋ ′ ( ξ ) ,$

where $ξ$ lies between $x$ and $y$ .

By virtue of the concavity of $ℋ$ , equation (8) yields

(9)

$ℋ ( x ) ≤ ℋ ( y ) + ( x − y ) ℋ ′ ( y ) .$

Since $ℋ$ is positive, (9) gives

(10)

$ℋ ( x ) ℋ ( y ) ≤ 1 + ( x − y ) ℋ ′ ( y ) ℋ ( y ) .$

Multiplying both sides of (10) by $( b − y ) α − 1 Γ ( α )$ and then integrating the resulting inequality with respect to $y$ over $[ a , b ]$ , we obtain

(11)

$ℋ ( x ) ∫ b a ( b − y ) α − 1 Γ ( α ) 1 ℋ ( y ) d y ≤ ∫ b a ( b − y ) α − 1 Γ ( α ) d y + ∫ b a ( b − y ) α − 1 Γ ( α ) ( x − y ) ℋ ′ ( y ) ℋ ( y ) d y = ( b − a ) α Γ ( α + 1 ) + ∫ b a ( b − y ) α − 1 Γ ( α ) ( ( b − y ) − ( b − x ) ) ℋ ′ ( y ) ℋ ( y ) d y = ( b − a ) α Γ ( α + 1 ) + 1 Γ ( α ) ∫ b a ( b − y ) α ℋ ′ ( y ) ℋ ( y ) d y − ( b − x ) Γ ( α ) ∫ b a ( b − y ) α − 1 ℋ ′ ( y ) ℋ ( y ) d y = ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ∫ b a 1 − y b α ℋ ′ ( y ) ℋ ( y ) d y − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − y b α − 1 ℋ ′ ( y ) ℋ ( y ) d y .$

We distinguish three cases: $0 < α ≤ 1 , 1 < α ≤ 2$ and $α > 2$ .

Assume that $0 < α ≤ 1$ , clearly $α − 1 ≤ 0$ and $a b ≤ y b ≤ 1$ because $y ∈ [ a , b ]$ . By using the Bernoulli inequality, we have

(12)

$1 − y b α ≤ 1 − α y b .$

By multiplying both sides of (12) by $b α Γ ( α ) ℋ ′ ( y ) ℋ ( y )$ , then integrating the result with respect to $y$ over $[ a , b ]$ , we obtain

(13)

$b α Γ ( α ) ∫ b a 1 − y b α ℋ ′ ( y ) ℋ ( y ) d y ≤ b α Γ ( α ) ∫ b a 1 − α y b ℋ ′ ( y ) ℋ ( y ) d y .$

On the other hand, we have $α − 1 ≤ 0$ . From Bernoulli inequality, we have

(14)

$1 − y b α − 1 ≥ 1 − ( α − 1 ) y b .$

By multiplying both sides of (14) by $− b α − 1 ( b − x ) Γ ( α ) ℋ ′ ( y ) ℋ ( y )$ , then integrating the result with respect to $y$ over $[ a , b ]$ , we obtain

(15)

$− b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − y b α − 1 ℋ ′ ( y ) ℋ ( y ) d y ≤ − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − ( α − 1 ) y b ℋ ′ ( y ) ℋ ( y ) d y .$

By using (13) and (15) in (11), then we obtain

(16)

$ℋ ( x ) ∫ b a ( b − y ) α − 1 Γ ( α ) 1 ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ∫ b a 1 − α y b ℋ ′ ( y ) ℋ ( y ) d y − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − ( α − 1 ) y b ℋ ′ ( y ) ℋ ( y ) d y = ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ( 1 − α ) ln ℋ ( b ) − ( 2 − α ) b α − 1 ( b − x ) Γ ( α ) ln ℋ ( b ) + b α − 2 ( b − x ) Γ ( α ) ( b − ( α − 1 ) a ) ln ℋ ( a ) − b α − 1 Γ ( α ) ( b − α a ) ln ℋ ( a ) + b α − 2 ( ( 2 − α ) b + ( 1 − α ) ( b − x ) ) Γ ( α ) ∫ b a ln ℋ ( y ) d y .$

Since $ln ℋ ( y )$ is concave, we have

(17)

$∫ b a ln ℋ ( y ) d y ≤ ( b − a ) ln ℋ a + b 2 .$

By using (17) in (16), we obtain

(18)

$ℋ ( x ) ∫ b a ( b − y ) α − 1 Γ ( α ) 1 ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ( 1 − α ) ln ℋ ( b ) − ( 2 − α ) b α − 1 ( b − x ) Γ ( α ) ln ℋ ( b ) + b α − 2 ( b − x ) Γ ( α ) ( b − ( α − 1 ) a ) ln ℋ ( a ) − b α − 1 Γ ( α ) ( b − α a ) ln ℋ ( a ) + b α − 2 ( ( 2 − α ) b + ( 1 − α ) ( b − x ) ) Γ ( α ) ( b − a ) ln ℋ a + b 2 .$

By multiplying both sides of (18) by $( Γ ( α + 1 ) ) 2 ( b − a ) 2 α ( b − x ) α − 1 Γ ( α )$ and then integrating the resulting inequality with respect to $x$ over $[ a , b ]$ , we obtain

(19)

$( Γ ( α + 1 ) ) 2 ( b − a ) 2 α J a + α ( ℋ ) ( b ) × J a + α ( h ∘ ℋ ) ( b ) ≤ 1 + α ( 1 − α ) b α ( b − a ) α − ( 2 − α ) α 2 b α − 1 ( α + 1 ) ( b − a ) α − 1 ln ℋ ( b ) + α 2 b α − 2 ( b − ( α − 1 ) a ) ( α + 1 ) ( b − a ) α − 1 − α b α − 1 ( b − α a ) ( b − a ) α ln ℋ ( a ) + b α − 1 ( 2 − α ) α ( b − a ) α + ( 1 − α ) α 2 b ( α + 1 ) ( b − a ) α − 1 ( b − a ) ln ℋ a + b 2 .$

Assume that $1 < α ≤ 2$ , from (11), we have

(20)

$ℋ ( x ) ∫ b a ( b − y ) α − 1 Γ ( α ) 1 ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ∫ b a 1 − y b α ℋ ′ ( y ) ℋ ( y ) d y − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − y b α − 1 ℋ ′ ( y ) ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ∫ b a 1 − y b ℋ ′ ( y ) ℋ ( y ) d y − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − y b ℋ ′ ( y ) ℋ ( y ) d y = ( b − a ) α Γ ( α + 1 ) + b α − 1 ( b − ( b − x ) ) b Γ ( α ) ( b − a ) ln ℋ ( a ) + b α − 1 ( b − ( b − x ) ) b Γ ( α ) ∫ b a ln ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α − 1 ( b − ( b − x ) ) b Γ ( α ) ( b − a ) ln ℋ ( a ) + b α − 1 ( b − ( b − x ) ) b Γ ( α ) ( b − a ) ln ℋ a + b 2 ,$

where we have used (17),

$1 − y b α ≤ 1 − y b for 1 < α ≤ 2$

and

$1 − y b α − 1 ≥ 1 − y b for 1 < α ≤ 2 .$

By multiplying both sides of (20) by $( Γ ( α + 1 ) ) 2 ( b − a ) 2 α ( b − x ) α − 1 Γ ( α )$ and then integrating the resulting inequality with respect to $x$ over $[ a , b ]$ , we obtain

$( Γ ( α + 1 ) ) 2 ( b − a ) 2 α J a + α ( ℋ ) ( b ) × J a + α ( h ∘ ℋ ) ( b ) ≤ 1 + ( α + 1 ) α b α − 1 − α 2 b α − 2 ( b − a ) ( α + 1 ) ( b − a ) α − 1 ln ℋ ( a ) ℋ a + b 2 .$

Assume that $α > 2$ , from (11), we have

(21)

$ℋ ( x ) ∫ b a ( b − y ) α − 1 Γ ( α ) 1 ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ∫ b a 1 − y b α ℋ ′ ( y ) ℋ ( y ) d y − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − y b α − 1 ℋ ′ ( y ) ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) + b α Γ ( α ) ∫ b a 1 − y b ℋ ′ ( y ) ℋ ( y ) d y − b α − 1 ( b − x ) Γ ( α ) ∫ b a 1 − ( α − 1 ) y b ℋ ′ ( y ) ℋ ( y ) d y = ( b − a ) α Γ ( α + 1 ) − b α − 1 ( b − a ) Γ ( α ) ln ℋ ( a ) + b α − 1 Γ ( α ) ∫ b a ln ℋ ( y ) d y − ( 2 − α ) b α − 1 ( b − x ) Γ ( α ) ln ℋ ( b ) + b α − 1 ( b − x ) b Γ ( α ) ( b − ( α − 1 ) a ) ln ℋ ( a ) − ( α − 1 ) b α − 1 ( b − x ) b Γ ( α ) ∫ b a ln ℋ ( y ) d y ,$

where we have used Bernoulli inequality and

$1 − y b α ≤ 1 − y b for α ≥ 1 .$

By using the fact that $ln ℋ ( y )$ is concave and $− ln ℋ ( y )$ is convex since $ℋ ( y )$ is concave, from (21), we obtain

(22)

$ℋ ( x ) ∫ b a ( b − y ) α − 1 Γ ( α ) 1 ℋ ( y ) d y ≤ ( b − a ) α Γ ( α + 1 ) − b α − 1 ( b − a ) Γ ( α ) ln ℋ ( a ) + b α − 1 Γ ( α ) ( b − a ) ln ℋ a + b 2 − ( 2 − α ) b α − 1 ( b − x ) Γ ( α ) ln ℋ ( b ) + b α − 1 ( b − x ) b Γ ( α ) ( b − ( α − 1 ) a ) ln ℋ ( a ) − ( α − 1 ) b α − 1 ( b − x ) 2 b Γ ( α ) ( b − a ) ln ℋ ( a ) − ( α − 1 ) b α − 1 ( b − x ) 2 b Γ ( α ) ( b − a ) ln ℋ ( b ) = ( b − a ) α Γ ( α + 1 ) + b α − 1 ( ( 3 − α ) b + ( 1 − α ) a ) 2 b Γ ( α ) ( b − x ) − b α − 1 ( b − a ) Γ ( α ) ln ℋ ( a ) − b α − 1 ( ( 3 − α ) b + ( 1 − α ) a ) 2 b Γ ( α ) ( b − x ) ln ℋ ( b ) + b α − 1 Γ ( α ) ( b − a ) ln ℋ a + b 2 .$

By multiplying both sides of (22) by $( Γ ( α + 1 ) ) 2 ( b − a ) 2 α ( b − x ) α − 1 Γ ( α )$ and then integrating the resulting inequality with respect to $x$ over $[ a , b ]$ , we obtain

$( Γ ( α + 1 ) ) 2 ( b − a ) 2 α J a + α ( ℋ ) ( b ) × J a + α ( h ∘ ℋ ) ( b ) ≤ 1 + α 2 b α − 1 ( ( 3 − α ) b + ( 1 − α ) a ) 2 ( α + 1 ) b ( b − a ) α − 1 ln ℋ ( a ) ℋ ( b ) + α b α − 1 ( b − a ) α − 1 ln ℋ a + b 2 ℋ ( a ) .$

The proof is over.□

Remark 2.4

By setting $α = 1$ , Theorem 2.3 will be reduced to Theorem 4.1 from [4].

3 Conclusion

We have successfully established a fractional version of the Schweitzer inequality tailored for the left-sided Riemann-Liouville fractional integral operator. Notably, for $α = 1$ , this fractional inequality reduces to the classical Schweitzer inequality, reinforcing its relevance and applicability. This development holds promise for diverse applications in the domain of fractional calculus.

Acknowledgement

T. Abdeljawad would like to thank Prince Sultan University for the support through TAS research lab.

Author contributions: All the authors thought of the study, contributed to its conception and coordination, prepared the manuscript, and reviewed and approved the final version.
Conflict of interest: The authors state no conflicts of interest.

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Received: 2023-10-16

Revised: 2023-12-27

Accepted: 2024-07-23

Published Online: 2024-08-26

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Articles in the same Issue

https://doi.org/10.1515/math-2024-0043

Keywords for this article

Schweitzer inequality; concave functions; Cauchy-Schwarz inequality; Bernoulli inequality; Riemann-Liouville integral operator

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