Home Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
Article Open Access

Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions

  • Huseyin Budak , Hasan Kara , Muhammad Aamir Ali , Sundas Khan and Yuming Chu EMAIL logo
Published/Copyright: September 21, 2021
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this work, we introduce the notions about the Riemann-Liouville fractional integrals for interval-valued functions on co-ordinates. We also establish Hermite-Hadamard and some related inequalities for co-ordinated convex interval-valued functions by applying the newly defined fractional integrals. The results of the present paper are the extension of several previously published results.

Keywords: fractional integrals; Hermite-Hadamard inequality; interval-valued functions
MSC 2010: 26D10; 26D15; 26A51

1 Introduction

The Hermite-Hadamard inequality discovered by Hermite and Hadamard (see, e.g., [1], [2, p. 137]) is one of the most well-established inequalities in the theory of convex functions having a geometrical interpretation and many applications. The Hermite-Hadamard inequality states that if f : I → R is a convex function on the interval I of real numbers and a , b ∈ I with a < b , then

(1) f a + b 2 ≤ 1 b − a ∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) 2 .

Both inequalities hold in the reversed direction if f is concave. We note that Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it can be easily done by using Jensen’s inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been studied (see, for example, [3,4,5, 6,7,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20, 21,22,23, 24,25,26] and references therein).

On the other hand, interval analysis is a particular case of set-valued analysis, which is the study of sets in the spirit of mathematical analysis and general topology. It was introduced as an attempt to handle the interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. An old example of interval enclosure is Archimede’s method which is related to computing of the circumference of a circle. In 1966, the first book related to interval analysis was given by Moore, who is known as the first user of intervals in computational mathematics [27]. After his book, several scientists started to investigate the theory and applications of interval arithmetic. Nowadays, because of its applications, interval analysis is a useful tool in various areas which are interested intensely in uncertain data. You can see applications in computer graphics, experimental and computational physics, error analysis, robotics, and many others.

In addition, several important inequalities (Hermite-Hadamard, Ostrowski, etc.) have been studied for interval-valued functions in recent years. In [28,29], Chalco-Cano et al. obtained Ostrowski-type inequalities for interval-valued functions by using Hukuhara derivative for interval-valued functions. However, inequalities were studied for more general set-valued maps. In [30,31,32], the authors proved different variants of the Hermite-Hadamard inequalities for interval-valued functions.

The purpose of this paper is to complete the Riemann-Liouville integrals for interval-valued functions and to obtain Hermite-Hadamard inequalities via these integrals. Furthermore, Hermite-Hadamard-type inequalities are given using these integrals.

2 Preliminaries

In this section, we recall some basic definitions, results, notions, and properties, which are used throughout the paper. We denote R ℐ + the family of all positive intervals of R . The Hausdorff distance between [ X ̲ , X ¯ ] and [ Y ̲ , Y ¯ ] is defined as

d ( [ X ̲ , X ¯ ] , [ Y ̲ , Y ¯ ] ) = max { ∣ X ̲ − Y ̲ ∣ , ∣ X ¯ − Y ¯ ∣ } .

The ( R ℐ , d ) is a complete metric space. For more details and basic notations on interval-valued functions, see [33,34].

In [27], Moore introduced the Riemann integral for interval-valued functions. The set of all Riemann integrable interval-valued functions and real-valued functions on [ a , b ] are denoted by ℐℛ ( [ a , b ] ) and ℛ ( [ a , b ] ) , respectively. The following theorem gives a relation between (IR)-integrable and Riemann integrable ( R -integrable) (see [33, p. 131]):

Theorem 1

Let F : [ a , b ] → R ℐ be an interval-valued function such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] . F ∈ ℐℛ ( [ a , b ] ) if and only if F ̲ ( t ) , F ¯ ( t ) ∈ ℛ ( [ a , b ] ) , and

( IR ) ∫ a b F ( t ) d t = ( R ) ∫ a b F ̲ ( t ) d t , ( R ) ∫ a b F ¯ ( t ) d t .

In [34,35], Zhao et al. introduced a class of convex interval-valued functions as follows:

Definition 1

Let h : J ⊂ R → R be a positive function. We say that F : I ⊂ R → R ℐ + is a h -convex interval-valued function, if for all x , y ∈ I and t ∈ ( 0 , 1 ) , we have

(2) h ( t ) F ( x ) + h ( 1 − t ) F ( y ) ⊆ F ( t x + ( 1 − t ) y ) .

With S X ( h , [ a , b ] , R ℐ + ) we will show the set of all h -convex interval-valued functions.

The usual notion of convex interval-valued function corresponds to relation (2) with h ( t ) = t , see [30]. Furthermore, if we take h ( t ) = t s in (2), then Definition 1 gives the convex interval-valued function defined by Breckner, see [36].

In [34], Zhao et al. obtained the following Hermite-Hadamard inequality for h -convex interval-valued functions:

Theorem 2

[34] Let F : [ a , b ] → R ℐ + be an interval-valued function such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] and F ∈ ℐℛ ( [ a , b ] ) , h : [ 0 , 1 ] → R be a non-negative function and h 1 2 ≠ 0 . If F ∈ S X ( h , [ a , b ] , R ℐ + ) , then

(3) 1 2 h 1 2 F a + b 2 ⊇ 1 b − a ( IR ) ∫ a b F ( x ) d x ⊇ [ F ( a ) + F ( b ) ] ∫ 0 1 h ( t ) d t .

Remark 1

  1. If h ( t ) = t , then (3) reduces to the following result:

    (4) F a + b 2 ⊇ 1 b − a ( IR ) ∫ a b F ( x ) d x ⊇ F ( a ) + F ( b ) 2 ,

    which is obtained by [30].

  2. If h ( t ) = t s , then (3) reduces to the following result:

    2 s − 1 F a + b 2 ⊇ 1 b − a ( IR ) ∫ a b F ( x ) d x ⊇ F ( a ) + F ( b ) s + 1 ,

    which is obtained by [37].

In [38], Budak et al. gave the fractional version of Hermite-Hadamard inequalities for convex interval-valued functions as follows:

Theorem 3

If F : [ a , b ] → R ℐ + is a convex interval-valued function such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] and α > 0 , then we have

(5) F a + b 2 ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b ) + J b − α F ( a ) ] ⊇ F ( a ) + F ( b ) 2 .

Theorem 4

If F , G : [ a , b ] → R ℐ + are two convex interval-valued functions such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] and G ( t ) = [ G ̲ ( t ) , G ¯ ( t ) ] , then for α > 0 we have

(6) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b ) G ( b ) + J b − α F ( a ) G ( a ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) M ( a , b ) + α ( α + 1 ) ( α + 2 ) N ( a , b )

and

(7) 2 F a + b 2 G a + b 2 ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b ) G ( b ) + J b − α F ( a ) G ( a ) ] + α ( α + 1 ) ( α + 2 ) M ( a , b ) + 1 2 − α ( α + 1 ) ( α + 2 ) N ( a , b ) ,

where

M ( a , b ) = F ( a ) G ( a ) + F ( b ) G ( b ) and N ( a , b ) = F ( a ) G ( b ) + F ( b ) G ( a ) .

3 Fractional integral of interval-valued functions

In this section, we introduce the notions of fractional double integral for interval-valued functions and recall some basic definitions of interval-valued integrals. We also give the definition of interval-valued convex functions on co-ordinates. In the sequel of the paper, Δ = [ a , b ] × [ c , d ] .

In [39], Lupulescu defined the following interval-valued left-sided Riemann-Liouville fractional integral.

Definition 2

Let F : [ a , b ] → R ℐ be an interval-valued function such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] and let α > 0 . The interval-valued left-sided Riemann-Liouville fractional integral of a function f is defined by

J a + α F ( x ) = 1 Γ ( α ) ( IR ) ∫ a x ( x − s ) α − 1 F ( t ) d t , x > a ,

where Γ is the Euler Gamma function.

Based on the definition of Lupulescu, Budak et al. in [38] gave the definition of interval-valued right-sided Riemann-Liouville fractional integral of the function by

J b − α F ( x ) = 1 Γ ( α ) ( IR ) ∫ x b ( s − x ) α − 1 F ( t ) d t , x < b ,

where Γ is the Euler Gamma function.

Theorem 5

If F : [ a , b ] → R ℐ is an interval-valued function such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] , then we have

J a + α F ¯ ( x ) = [ I a + α F ̲ ( x ) , I a + α F ¯ ( x ) ]

and

J b − α f ( x ) = [ I b − α F ̲ ( x ) , I b − α F ¯ ( x ) ] .

Now we recall the concept of interval-valued double integral given by Zhao et al. in [40]:

Theorem 6

[40] Let F : Δ = [ a , b ] × [ c , d ] → R ℐ . Then F is called ID-integrable on Δ with ID-integral U = ( ID ) ∬ Δ F ( t , s ) d A , if for any ε > 0 there exists δ > 0 such that

d ( S ( F , P , δ , Δ ) ) < ε

for any P ∈ P ( δ , Δ ) . The collection of all ID-integrable functions on Δ will be denoted by ℐ D ( Δ ) . For more details about the notations used here, one can read [40].

Theorem 7

[40] Let Δ = [ a , b ] × [ c , d ] . If F : Δ → R ℐ is ID-integrable on Δ , then we have

( ID ) ∬ Δ F ( s , t ) d A = ( IR ) ∫ a b ( IR ) ∫ c d F ( s , t ) d s d t .

Example 1

Let F : Δ = [ 0 , 1 ] × [ 1 , 2 ] → R ℐ + be defined by

F ( s , t ) = [ s t , s + t ] ,

then F ( s , t ) is integrable on Δ and ( ID ) ∬ Δ F ( s , t ) d A = 3 4 , 2 .

By applying the concepts of Lupulescu [39] and Zhao et al. [40] about interval-valued integrals, we can define interval-valued Riemann-Liouville double fractional integral of the function F ( x , y ) by

Definition 3

Let F ∈ L 1 ( [ a , b ] × [ c , d ] ) . The Riemann-Liouville integrals J a + , c + α , β , J a + , d − α , β , + J b − , c + α , β , and J b − , d − α , β of order α , β > 0 with a , c ≥ 0 are defined by

J a + , c + α , β F ( x , y ) = 1 Γ ( α ) Γ ( β ) ( IR ) ∫ a x ∫ c y ( x − t ) α − 1 ( y − s ) β − 1 F ( t , s ) d s d t , x > a , y > c , J a + , d − α , β F ( x , y ) = 1 Γ ( α ) Γ ( β ) ( IR ) ∫ a x ∫ y d ( x − t ) α − 1 ( s − y ) β − 1 F ( t , s ) d s d t , x > a , y > d , J b − , c + α , β F ( x , y ) = 1 Γ ( α ) Γ ( β ) ( IR ) ∫ x b ∫ c y ( t − x ) α − 1 ( y − s ) β − 1 F ( t , s ) d s d t , x < b , y > c , J b − , d − α , β F ( x , y ) = 1 Γ ( α ) Γ ( β ) ( IR ) ∫ x b ∫ y d ( t − x ) α − 1 ( s − y ) β − 1 F ( t , s ) d s d t , x < b , y < d ,

respectively.

Definition 4

[41] A function F : Δ → R ℐ + is said to be interval-valued co-ordinated convex function, if the following inequality holds

F ( t x + ( 1 − t ) y , s u + ( 1 − s ) w ) ⊇ t s F ( x , u ) + t ( 1 − s ) F ( x , w ) + s ( 1 − t ) F ( y , u ) + ( 1 − s ) ( 1 − t ) F ( y , w ) ,

for all ( x , y ) , ( u , w ) ∈ Δ and s , t ∈ [ 0 , 1 ] .

Lemma 1

[41] A function F : Δ → R ℐ + is interval-valued convex on co-ordinates if and only if there exist two functions F x : [ c , d ] → R ℐ + , F x ( w ) = F ( x , w ) and F y : [ a , b ] → R ℐ + , F y ( u ) = F ( y , u ) are interval-valued convex.

It is easy to prove that an interval-valued convex function is interval-valued co-ordinated convex but the converse may not be true. For this we can see the following example.

Example 2

An interval-valued function f : [ 0 , 1 ] 2 → R ℐ + defined as F ( x , y ) = [ x y , ( 6 − e x ) ( 6 − e y ) ] is interval-valued convex on co-ordinates but is not interval-valued convex on [ 0 , 1 ] 2 .

Proposition 1

[41] If F , G : Δ → R ℐ + are two interval-valued co-ordinated convex functions on Δ and α ≥ 0 , then F + G and α F are interval-valued co-ordinated convex functions.

Proposition 2

[41] If F , G : Δ → R ℐ + are two interval-valued co-ordinated convex functions on Δ , then ( F G ) is interval-valued co-ordinated convex function on Δ .

4 Main results

In this section, we establish Hermite-Hadamard integral inequalities for interval-valued co-ordinated convex functions by applying interval-valued double fractional integral. We also present inequalities of Hermite-Hadamard-type for the product of interval-valued co-ordinated convex functions.

Theorem 8

If F : Δ → R ℐ + is an interval-valued co-ordinated convex function on Δ such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] , then the following inequalities hold:

(8) F a + b 2 , c + d 2 ⊇ Γ ( α + 1 ) 4 ( b − a ) α J a + α F b , c + d 2 + J b − α F a , c + d 2 + Γ ( β + 1 ) 4 ( d − c ) β J c + β F a + b 2 , d + J d − β F a + b 2 , c ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) + J a + , d − α , β F ( b , c ) + J b − , c + α , β F ( a , d ) + J b − , d − α , β F ( a , c ) ] ⊇ Γ ( α + 1 ) 8 ( b − a ) α [ J a + α F ( b , c ) + J a + α F ( b , d ) + J b − α F ( a , c ) + J b − α F ( a , d ) ] + Γ ( β + 1 ) 4 ( d − c ) β [ J c + β F ( a , d ) + J c + β F ( b , d ) + J d − β F ( a , c ) + J d − β F ( b , c ) ] ⊇ F ( a , c ) + F ( a , d ) + F ( b , c ) + F ( b , d ) 4 .

Proof

Since F is an interval-valued co-ordinated convex function on Δ , it follows that the mapping, G y : [ a , b ] → R ℐ + , G y ≔ F ( x , y ) is an interval-valued convex on [ a , b ] for all x ∈ [ a , b ] . From inequality (5), we have:

G y a + b 2 ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α G y ( b ) + J b − α G y ( a ) ] ⊇ G y ( a ) + G y ( b ) 2 .

That can be written as,

F a + b 2 , y ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , y ) + J b − α F ( a , y ) ] ⊇ F ( a , y ) + F ( b , y ) 2 .

That is,

(9) F a + b 2 , y ⊇ α 2 ( b − a ) α ( IR ) ∫ a b ( b − t ) α − 1 F ( t , y ) d t + ( IR ) ∫ a b ( t − a ) α − 1 F ( t , y ) d t ⊇ F ( a , y ) + F ( b , y ) 2 .

Multiplying the both sides of inequality (28) by β ( d − y ) β − 1 2 ( d − c ) β and integrating the resultant one with respect to y over [ c , d ] , we have

(10) β 2 ( d − c ) β ( IR ) ∫ c d F a + b 2 , y ( d − y ) β − 1 d y ⊇ ( IR ) ∫ a b ∫ c d ( b − t ) α − 1 ( d − y ) β − 1 F ( t , y ) d y d t + ( IR ) ∫ a b ∫ c d ( t − a ) α − 1 ( d − y ) β − 1 F ( t , y ) d y d t ⊇ 1 2 β 2 ( d − c ) β ( IR ) ∫ c d F ( a , y ) ( d − y ) β − 1 d y + β 2 ( d − c ) β ( IR ) ∫ c d F ( b , y ) ( d − y ) β − 1 d y .

Again, multiplying the both sides of inequality (28) by β ( y − c ) β − 1 2 ( d − c ) β and integrating the resultant one with respect to y over [ c , d ] , we have

(11) β 2 ( d − c ) β ( IR ) ∫ c d F a + b 2 , y ( y − c ) β − 1 d y ⊇ α β 4 ( b − a ) α ( d − c ) β ( IR ) ∫ a b ∫ c d ( b − t ) α − 1 ( y − c ) β − 1 F ( t , y ) d y d t + ( IR ) ∫ a b ∫ c d ( t − a ) α − 1 ( y − c ) β − 1 F ( t , y ) d y d t

⊇ 1 2 β 2 ( d − c ) β ( IR ) ∫ c d F ( a , y ) ( y − c ) β − 1 d y + β 2 ( d − c ) β ( IR ) ∫ c d F ( b , y ) ( y − c ) β − 1 d y .

Moreover, inequality (10) can be written as

(12) Γ ( β + 1 ) 2 ( d − c ) β J c + β F a + b 2 , d ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) + J b − , c + α , β F ( a , d ) ] ⊇ Γ ( β + 1 ) 4 ( d − c ) β J c + β F ( a , d ) + J c + β F ( b , d )

and inequality (11) can be written as

(13) Γ ( β + 1 ) 2 ( d − c ) β J d − β F a + b 2 , c ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , d − α , β F ( b , c ) + J b − , d − α , β F ( a , c ) ] ⊇ Γ ( β + 1 ) 4 ( d − c ) β J d − β F ( a , c ) + J d − β F ( b , c ) .

Similarly, H x : [ c , d ] → R ℐ + , H x ( y ) ≔ F ( x , y ) is an interval-valued convex function on [ c , d ] and y ∈ [ a , b ] , we get

(14) Γ ( α + 1 ) 2 ( b − a ) α J a + α F b , c + d 2 ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) + J a + , d − α , β F ( b , c ) ] ⊇ Γ ( α + 1 ) 4 ( b − a ) α J a + α F ( b , c ) + J a + α F ( b , d )

and

(15) Γ ( α + 1 ) 2 ( b − a ) α J b − α F a , c + d 2 ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J b − , c + α , β F ( a , d ) + J b − , d − α , β F ( a , c ) ] ⊇ Γ ( α + 1 ) 4 ( b − a ) α [ J b − α F ( a , c ) + J b − α F ( a , d ) ] .

Summing inequalities (12), (13), (14), and (15), we obtain second, third, and fourth inequalities of Theorem 8.

From (5), we have

(16) F a + b 2 , c + d 2 ⊇ Γ ( α + 1 ) 2 ( b − a ) α J a + α F b , c + d 2 + J b − α F a , c + d 2

(17) F a + b 2 , c + d 2 ⊇ Γ ( β + 1 ) 2 ( d − c ) β J c + β F a + b 2 , d + J d − β F a + b 2 , c .

By adding (16) and (17) and using Theorem 9, we have the first inequality in (8),

F a + b 2 , c + d 2 ⊇ Γ ( α + 1 ) 4 ( b − a ) α J a + α F b , c + d 2 + J b − α F a , c + d 2 + Γ ( β + 1 ) 4 ( d − c ) β J c + β F a + b 2 , d + J d − β F a + b 2 , c .

At the end, again from (3) and Theorem 9, we have

(18) Γ ( α + 1 ) 2 ( b − a ) α J a + α F ( b , c ) + J b − α F ( a , c ) ⊇ F ( a , c ) + F ( b , c ) 2 ,

(19) Γ ( α + 1 ) 2 ( b − a ) α J a + α F ( b , d ) + J b − α F ( a , d ) ⊇ F ( a , d ) + F ( b , d ) 2 ,

(20) Γ ( β + 1 ) 2 ( d − c ) β J c + β F ( a , d ) + J d − β F ( a , c ) ⊇ F ( a , c ) + F ( a , d ) 2 ,

(21) Γ ( β + 1 ) 2 ( d − c ) β J c + β F ( b , d ) + J d − β F ( b , c ) ⊇ F ( b , c ) + F ( b , d ) 2 .

If we add (18), (19), (20), and (21) and then multiplying by 1 4 , we have the last inequality of Theorem 8.

Γ ( α + 1 ) 8 ( b − a ) α [ J a + α F ( b , c ) + J a + α F ( b , d ) + J b − α F ( a , c ) + J b − α F ( a , d ) ] + Γ ( β + 1 ) 4 ( d − c ) β [ J c + β F ( a , d ) + J c + β F ( b , d ) + J d − β F ( a , c ) + J d − β F ( b , c ) ] ⊇ F ( a , c ) + F ( a , d ) + F ( b , c ) + F ( b , d ) 4 ,

and the proof is completed.□

Remark 2

If we choose α = β = 1 in Theorem 8, then we have

F a + b 2 , c + d 2 ⊇ 1 2 1 b − a ( IR ) ∫ a b F x , c + d 2 d x + 1 d − c ( IR ) ∫ c d F a + b 2 , y d y ⊇ 1 ( b − a ) ( d − c ) ( IR ) ∫ a b ∫ c d F ( x , y ) d y d x ⊇ 1 4 1 b − a ( IR ) ∫ a b [ F ( x , c ) + F ( x , d ) ] d x + 1 d − c ( IR ) ∫ c d [ F ( a , y ) + F ( b , y ) ] d y ⊇ F ( a , c ) + F ( a , d ) + F ( b , c ) + F ( b , d ) 4 ,

which is proved by Zhao et al. in [41].

Remark 3

If F ̲ ( t ) = F ¯ ( t ) in Theorem 8, then Theorem 8 reduces to the result of Sarikaya [42, Theorem 4].

Theorem 9

If F , G : Δ → R ℐ + are two interval-valued co-ordinated convex functions on Δ such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] and G ( t ) = [ G ̲ ( t ) , G ¯ ( t ) ] , then the following inequality holds:

(22) Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β × [ J a + , c + α , β F ( b , d ) G ( b , d ) + J a + , d − α , β F ( b , c ) G ( b , c ) + J b − , c + α , β F ( a , d ) G ( a , d ) + J b − , d − α , β F ( a , c ) G ( a , c ) ] ⊇ 1 2 − β ( β + 1 ) ( β + 2 ) 1 2 − α ( α + 1 ) ( α + 2 ) K ( a , b , c , d ) + 1 2 − β ( β + 1 ) ( β + 2 ) α ( α + 1 ) ( α + 2 ) L ( a , b , c , d ) + β ( β + 1 ) ( β + 2 ) 1 2 − α ( α + 1 ) ( α + 2 ) M ( a , b , c , d ) + β ( β + 1 ) ( β + 2 ) α ( α + 1 ) ( α + 2 ) N ( a , b , c , d ) ,

where

K ( a , b , c , d ) = F ( a , c ) G ( a , c ) + F ( b , c ) G ( b , c ) + F ( a , d ) G ( a , d ) + F ( b , d ) G ( b , d ) ,

L ( a , b , c , d ) = F ( a , c ) G ( b , c ) + F ( b , c ) G ( a , c ) + F ( a , d ) G ( b , d ) + F ( b , d ) G ( a , d ) , M ( a , b , c , d ) = F ( a , c ) G ( a , d ) + F ( b , c ) G ( b , d ) + F ( a , d ) G ( a , c ) + F ( b , d ) G ( b , c ) ,

and

N ( a , b , c , d ) = F ( a , c ) G ( b , d ) + F ( b , c ) G ( a , d ) + F ( a , d ) G ( b , c ) + F ( b , d ) G ( a , c ) .

Proof

Since F and G are interval-valued co-ordinated convex functions on Δ , if we define the mappings F x : [ c , d ] → R ℐ + , F x ( y ) = F ( x , y ) , and G x : [ c , d ] → R ℐ + , G x ( y ) = G ( x , y ) , then F x ( y ) and G x ( y ) are convex functions on [ c , d ] for all x ∈ [ a , b ] . If we apply inequality (6) for the convex functions F x ( y ) and G x ( y ) , then we have

(23) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F x ( d ) G x ( d ) + J d − β F x ( c ) G x ( c ) ] ⊇ 1 2 − β ( β + 1 ) ( β + 2 ) [ F x ( c ) G x ( c ) + F x ( d ) G x ( d ) ] + β ( β + 1 ) ( β + 2 ) [ F x ( c ) G x ( d ) + F x ( d ) G x ( c ) ] .

That is,

(24) β 2 ( d − c ) β ( IR ) ∫ c d ( d − y ) β − 1 F ( x , y ) G ( x , y ) d y + ( IR ) ∫ c d ( y − a ) β − 1 F ( x , y ) G ( x , y ) d y ⊇ 1 2 − β ( β + 1 ) ( β + 2 ) [ F ( x , c ) G ( x , c ) + F ( x , d ) G ( x , d ) ] + β ( β + 1 ) ( β + 2 ) [ F ( x , c ) G ( x , d ) + F ( x , d ) G ( x , c ) ] .

Multiplying inequality (24) by α 2 ( b − a ) α ( b − x ) α − 1 and integrating the resultant one with respect to x over [ a , b ] , we obtain

(25) Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) G ( b , d ) + J a + , d − α , β F ( b , c ) G ( b , c ) ] ⊇ Γ ( α + 1 ) 2 ( b − a ) α 1 2 − β ( β + 1 ) ( β + 2 ) [ J a + α F ( b , c ) G ( b , c ) + J a + α F ( b , d ) G ( b , d ) ] + Γ ( α + 1 ) 2 ( b − a ) α β ( β + 1 ) ( β + 2 ) [ J a + α F ( b , c ) G ( b , d ) + J a + α F ( b , d ) G ( b , c ) ] .

Similarly, multiplying inequality (24) by α 2 ( b − a ) α ( x − a ) α − 1 and integrating the resultant one with respect to x on [ a , b ] , we have

(26) Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J b − , c + α , β F ( a , d ) G ( a , d ) + J b − , d − α , β F ( a , c ) G ( a , c ) ] ⊇ Γ ( α + 1 ) 2 ( b − a ) α 1 2 − β ( β + 1 ) ( β + 2 ) [ J b − α F ( a , c ) G ( a , c ) + J b − α F ( a , d ) G ( a , d ) ] + Γ ( α + 1 ) 2 ( b − a ) α β ( β + 1 ) ( β + 2 ) [ J b − α F ( a , c ) G ( a , d ) + J b − α F ( a , d ) g ( a , c ) ] .

From (25) and (26), we get

(27) Γ ( α + 1 ) Γ ( β + 1 ) 2 ( b − a ) α ( d − c ) β × [ J a + , c + α , β f ( b , d ) g ( b , d ) + J a + , d − α , β F ( b , c ) G ( b , c ) + J b − , c + α , β F ( a , d ) G ( a , d ) + J b − , d − α , β F ( a , c ) G ( a , c ) ]

⊇ Γ ( α + 1 ) 2 ( b − a ) α 1 2 − β ( β + 1 ) ( β + 2 ) [ J a + α F ( b , c ) G ( b , c ) + J b − α F ( a , c ) G ( a , c ) ] + Γ ( α + 1 ) 2 ( b − a ) α 1 2 − β ( β + 1 ) ( β + 2 ) [ J a + α F ( b , d ) G ( b , d ) + J b − α F ( a , d ) G ( a , d ) ] + Γ ( α + 1 ) 2 ( b − a ) α β ( β + 1 ) ( β + 2 ) [ J a + α F ( b , c ) G ( b , d ) + J b − α F ( a , c ) G ( a , d ) ] + Γ ( α + 1 ) 2 ( b − a ) α β ( β + 1 ) ( β + 2 ) [ J a + α F ( b , d ) G ( b , c ) + J b − α F ( a , d ) G ( a , c ) ] .

For each term of the right hand side of (27), by inequality (6), we have

(28) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , c ) + J b − α F ( a , c ) G ( a , c ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( a , c ) + F ( b , c ) G ( b , c ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( b , c ) + F ( b , c ) G ( a , c ) ] ,

(29) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , d ) G ( b , d ) + J b − α F ( a , d ) G ( a , d ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( a , d ) + F ( b , d ) G ( b , d ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( b , d ) + F ( b , d ) G ( a , d ) ] ,

(30) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , d ) + J b − α F ( a , c ) G ( a , d ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( a , d ) + F ( b , c ) G ( b , d ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( b , d ) + F ( b , c ) G ( a , d ) ] ,

and

(31) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , d ) G ( b , c ) + J b − α F ( a , d ) G ( a , c ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( a , c ) + F ( b , d ) G ( b , c ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( b , c ) + F ( b , d ) G ( a , c ) ] .

If we substitute (28)–(31) in (27), we obtain the desired result (22).□

Remark 4

If we choose α = β = 1 in Theorem 9, then we have

1 ( b − a ) ( d − c ) ∫ a b ∫ c d f ( t , s ) d s d t ⊇ 1 9 K ( a , b , c , d ) + 1 18 [ L ( a , b , c , d ) + M ( a , b , c , d ) ] + 1 36 N ( a , b , c , d ) ,

which is proved by Zhao et al. in [41].

Corollary 1

If we choose G ( x , y ) = [ 1 , 1 ] for all ( x , y ) ∈ Δ in Theorem 9, then we have the following inequality:

Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) + J a + , d − α , β F ( b , c ) + J b − , c + α , β F ( a , d ) + J b − , d − α , β F ( a , c ) ] ⊇ 1 4 [ F ( a , c ) + F ( b , c ) + F ( a , d ) + F ( b , d ) ] .

Remark 5

If F ̲ ( t ) = F ¯ ( t ) in Theorem 9, then Theorem 9 reduces to the result of Budak and Sarikaya [43, Theorem 2.1].

Theorem 10

If F , G : Δ → R ℐ + are two interval-valued co-ordinated convex functions on Δ such that F ( t ) = [ F ̲ ( t ) , F ¯ ( t ) ] and G ( t ) = [ G ̲ ( t ) , G ¯ ( t ) ] , then the following Hermite-Hadamard-type inequality holds

(32) 4 F a + b 2 , c + d 2 G a + b 2 , c + d 2 ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β × [ J a + , c + α , β F ( b , d ) G ( b , d ) + J a + , d − α , β F ( b , c ) G ( b , c ) + J b − , c + α , β F ( a , d ) G ( a , d ) + J b − , d − α , β F ( a , c ) G ( a , c ) ] + α 2 ( α + 1 ) ( α + 2 ) + β ( β + 1 ) ( β + 2 ) 1 2 − α ( α + 1 ) ( α + 2 ) K ( a , b , c , d ) + 1 2 1 2 − α ( α + 1 ) ( α + 2 ) + α ( α + 1 ) ( α + 2 ) β ( β + 1 ) ( β + 2 ) L ( a , b , c , d ) + 1 2 1 2 − β ( β + 1 ) ( β + 2 ) + α ( α + 1 ) ( α + 2 ) β ( β + 1 ) ( β + 2 ) M ( a , b , c , d ) + 1 4 − α ( α + 1 ) ( α + 2 ) β ( β + 1 ) ( β + 2 ) N ( a , b , c , d ) ,

where K ( a , b , c , d ) , L ( a , b , c , d ) , M ( a , b , c , d ) , and N ( a , b , c , d ) are defined as in Theorem 9.

Proof

Since F and G are interval-valued co-ordinated convex functions on Δ , by inequality (7), we have

(33) 2 F a + b 2 , c + d 2 G a + b 2 , c + d 2 ⊇ α 2 ( b − a ) α ( IR ) ∫ a b ( b − x ) α − 1 F x , c + d 2 G x , c + d 2 d x + ( IR ) ∫ a b ( x − a ) α − 1 F x , c + d 2 G x , c + d 2 d x + α ( α + 1 ) ( α + 2 ) F a , c + d 2 G a , c + d 2 + F b , c + d 2 G b , c + d 2 + 1 2 − α ( α + 1 ) ( α + 2 ) F a , c + d 2 G b , c + d 2 + F b , c + d 2 G a , c + d 2

and

(34) 2 F a + b 2 , c + d 2 G a + b 2 , c + d 2 ⊇ β 2 ( d − c ) β ( IR ) ∫ c d ( d − y ) β − 1 F a + b 2 , y G a + b 2 , y d y + ( IR ) ∫ c d ( y − c ) β − 1 F a + b 2 , y G a + b 2 , y d y + β ( β + 1 ) ( β + 2 ) F a + b 2 , c G a + b 2 , c + F a + b 2 , d G a + b 2 , d + 1 2 − β ( β + 1 ) ( β + 2 ) F a + b 2 , c G a + b 2 , d + F a + b 2 , d G a + b 2 , c .

From (33) and (34), we get

(35) 8 F a + b 2 , c + d 2 G a + b 2 , c + d 2 ⊇ α 2 ( b − a ) α ∫ a b ( b − x ) α − 1 2 F x , c + d 2 G x , c + d 2 d x + ( IR ) ∫ a b ( x − a ) α − 1 2 F x , c + d 2 G x , c + d 2 d x

× β 2 ( d − c ) β ( IR ) ∫ c d ( d − y ) β − 1 2 F a + b 2 , y G a + b 2 , y d y + ∫ c d ( y − c ) β − 1 2 F a + b 2 , y G a + b 2 , y d y + α ( α + 1 ) ( α + 2 ) 2 F a , c + d 2 G a , c + d 2 + 2 F b , c + d 2 G b , c + d 2 + 1 2 − α ( α + 1 ) ( α + 2 ) 2 F a , c + d 2 G b , c + d 2 + 2 F b , c + d 2 G a , c + d 2 + β ( β + 1 ) ( β + 2 ) 2 F a + b 2 , c G a + b 2 , c + 2 F a + b 2 , d G a + b 2 , d + 1 2 − β ( β + 1 ) ( β + 2 ) 2 F a + b 2 , c G a + b 2 , d + 2 F a + b 2 , d G a + b 2 , c .

Since the mappings F x : [ c , d ] → R ℐ + , F x ( y ) = F ( x , y ) and G x : [ c , d ] → R ℐ + , G x ( y ) = G ( x , y ) are convex interval-valued, by applying inequality (7), we have

(36) 2 F a , c + d 2 G a , c + d 2 ⊇ Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( a , d ) + J d − β F ( a , c ) G ( a , c ) ] + β ( β + 1 ) ( β + 2 ) [ F ( a , c ) G ( a , c ) + F ( a , d ) G ( a , d ) ] + 1 2 − β ( β + 1 ) ( β + 2 ) [ F ( a , c ) G ( a , d ) + F ( a , d ) G ( a , c ) ] ,

(37) 2 F b , c + d 2 G b , c + d 2 ⊇ Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( b , d ) G ( b , d ) + J d − β F ( b , c ) G ( b , c ) ] + β ( β + 1 ) ( β + 2 ) [ F ( b , c ) G ( b , c ) + F ( b , d ) G ( b , d ) ] + 1 2 − β ( β + 1 ) ( β + 2 ) [ F ( b , c ) G ( b , d ) + F ( b , d ) G ( b , c ) ] ,

(38) 2 F a , c + d 2 G b , c + d 2 ⊇ Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( b , d ) + J d − β F ( a , c ) G ( b , c ) ] + β ( β + 1 ) ( β + 2 ) [ F ( a , c ) G ( b , c ) + F ( a , d ) G ( b , d ) ] + 1 2 − β ( β + 1 ) ( β + 2 ) [ F ( a , c ) G ( b , d ) + F ( a , d ) G ( b , c ) ] ,

and

(39) 2 F b , c + d 2 G a , c + d 2 ⊇ Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( b , d ) G ( a , d ) + J d − β F ( b , c ) G ( a , c ) ] + β ( β + 1 ) ( β + 2 ) [ F ( b , c ) G ( a , c ) + F ( b , d ) G ( a , d ) ] + 1 2 − β ( β + 1 ) ( β + 2 ) [ F ( b , c ) G ( a , d ) + F ( b , d ) G ( a , c ) ] .

Similarly, since the mappings F y : [ a , b ] → R ℐ + , F y ( x ) = F ( x , y ) , and G y : [ a , b ] → R ℐ + , G y ( x ) = G ( x , y ) are convex interval-valued, by applying inequality (7), we have

(40) 2 F a + b 2 , c G a + b 2 , c ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , c ) + J b − α F ( a , c ) G ( a , c ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( a , c ) + F ( b , c ) G ( b , c ) ] + 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( b , c ) + F ( b , c ) G ( a , c ) ] ,

(41) 2 F a + b 2 , d G a + b 2 , d ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , d ) G ( b , d ) + J b − α F ( a , d ) G ( a , d ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( a , d ) + F ( b , d ) G ( b , d ) ] + 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( b , d ) + F ( b , d ) G ( a , d ) ] ,

(42) 2 F a + b 2 , c G a + b 2 , d ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , d ) + J b − α F ( a , c ) G ( a , d ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( a , d ) + F ( b , c ) G ( b , d ) ] + 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , c ) G ( b , d ) + F ( b , c ) G ( a , d ) ]

and

(43) 2 F a + b 2 , d G a + b 2 , c ⊇ Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , d ) G ( b , c ) + J b − α F ( a , d ) G ( a , c ) ] + α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( a , c ) + F ( b , d ) G ( b , c ) ] + 1 2 − α ( α + 1 ) ( α + 2 ) [ F ( a , d ) G ( b , c ) + F ( b , d ) G ( a , c ) ] .

On the other hand, by applying inequality (7), we get

(44) α 2 ( b − a ) α ( IR ) ∫ a b ( b − x ) α − 1 2 F x , c + d 2 G x , c + d 2 d x ⊇ α β 4 ( b − a ) α ( d − c ) β ( IR ) ∫ a b ∫ c d ( b − x ) α − 1 ( d − y ) β − 1 F ( x , y ) G ( x , y ) d y d x + ( IR ) ∫ a b ∫ c d ( b − x ) α − 1 ( y − c ) β − 1 F ( x , y ) G ( x , y ) d y d x + β ( β + 1 ) ( β + 2 ) α 2 ( b − a ) α ( IR ) ∫ a b ( b − x ) α − 1 [ F ( x , c ) G ( x , c ) + F ( x , d ) G ( x , d ) ] d x + 1 2 − β ( β + 1 ) ( β + 2 ) α 2 ( b − a ) α ( IR ) ∫ a b ( b − x ) α − 1 [ F ( x , c ) G ( x , d ) + F ( x , d ) G ( x , c ) ] d x = Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) G ( b , d ) + J a + , d − α , β F ( b , c ) G ( b , c ) ] + β ( β + 1 ) ( β + 2 ) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , c ) + J a + α F ( b , d ) G ( b , d ) ] + 1 2 − β ( β + 1 ) ( β + 2 ) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , d ) + J a + α F ( b , d ) G ( b , c ) ] .

Similarly, we also have

(45) α 2 ( b − a ) α ( IR ) ∫ a b ( x − a ) α − 1 2 F x , c + d 2 G x , c + d 2 d x ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J b − , c + α , β F ( a , d ) G ( a , d ) + J b − , d − α , β F ( a , c ) G ( a , c ) ]

+ β ( β + 1 ) ( β + 2 ) Γ ( α + 1 ) 2 ( b − a ) α [ J b − α F ( a , c ) G ( a , c ) + J b − α F ( a , d ) G ( a , d ) ] + 1 2 − β ( β + 1 ) ( β + 2 ) Γ ( α + 1 ) 2 ( b − a ) α [ J b − α F ( a , c ) G ( a , d ) + J b − α F ( a , d ) G ( a , c ) ] ,

(46) β 2 ( d − c ) β ( IR ) ∫ c d ( d − y ) β − 1 2 F a + b 2 , y G a + b 2 , y d y ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) G ( b , d ) + J b − , c + α , β F ( a , d ) G ( a , d ) ] + α ( α + 1 ) ( α + 2 ) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( a , d ) + J c + β F ( b , d ) G ( b , d ) ] + 1 2 − α ( α + 1 ) ( α + 2 ) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( b , d ) + J c + β F ( b , d ) G ( a , d ) ] ,

and

(47) β 2 ( d − c ) β ∫ c d ( y − c ) β − 1 2 F a + b 2 , y G a + b 2 , y d y ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , d − α , β F ( b , c ) G ( b , c ) + J b − , d − α , β F ( a , c ) G ( a , c ) ] + α ( α + 1 ) ( α + 2 ) Γ ( β + 1 ) 2 ( d − c ) β [ J d − β F ( a , c ) G ( a , c ) + J d − β F ( b , c ) G ( b , c ) ] + 1 2 − α ( α + 1 ) ( α + 2 ) Γ ( β + 1 ) 2 ( d − c ) β [ J d − β F ( a , c ) G ( b , c ) + J d − β F ( b , c ) G ( a , c ) ] .

If we substitute (36)–(47) in (35), we get

(48) 8 F a + b 2 , c + d 2 G a + b 2 , c + d 2 ⊇ Γ ( α + 1 ) Γ ( β + 1 ) ( 2 b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) G ( b , d ) + J a + , d − α , β F ( b , c ) G ( b , c ) + J b − , c + α , β F ( a , d ) G ( a , d ) + J b − , d − α , β F ( a , c ) G ( a , c ) ] + 2 α ( α + 1 ) ( α + 2 ) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( a , d ) + J c + β F ( b , d ) G ( b , d ) ] + Γ ( β + 1 ) 2 ( d − c ) β [ J d − β F ( a , c ) G ( a , c ) + J d − β F ( b , c ) G ( b , c ) ] + 2 1 2 − α ( α + 1 ) ( α + 2 ) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( b , d ) + J c + β F ( b , d ) G ( a , d ) ] + Γ ( β + 1 ) 2 ( d − c ) β [ J d − β F ( a , c ) G ( b , c ) + J d − β F ( b , c ) G ( a , c ) ] + 2 β ( β + 1 ) ( β + 2 ) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , c ) + J a + α F ( b , d ) G ( b , d ) ] + Γ ( α + 1 ) 2 ( b − a ) α [ J b − α F ( a , c ) G ( a , c ) + J b − α F ( a , d ) G ( a , d ) ] + 2 1 2 − β ( β + 1 ) ( β + 2 ) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , d ) + J a + α F ( b , d ) G ( b , c ) ] + Γ ( α + 1 ) 2 ( b − a ) α [ J b − α F ( a , c ) G ( a , d ) + J b − α F ( a , d ) G ( a , c ) ]

+ 2 α ( α + 1 ) ( α + 2 ) β ( β + 1 ) ( β + 2 ) K ( a , b , c , d ) + 2 α ( α + 1 ) ( α + 2 ) 1 2 − β ( β + 1 ) ( β + 2 ) M ( a , b , c , d ) + 1 2 − α ( α + 1 ) ( α + 2 ) 2 β ( β + 1 ) ( β + 2 ) L ( a , b , c , d ) + 2 1 2 − α ( α + 1 ) ( α + 2 ) 1 2 − β ( β + 1 ) ( β + 2 ) N ( a , b , c , d ) .

Using inequality (6), we have

(49) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( a , d ) + J c + β F ( b , d ) G ( b , d ) ] + Γ ( β + 1 ) 2 ( d − c ) β [ J d − β F ( a , c ) G ( a , c ) + J d − β F ( b , c ) G ( b , c ) ] ⊇ 1 2 − β ( β + 1 ) ( β + 2 ) K ( a , b , c , d ) + β ( β + 1 ) ( β + 2 ) M ( a , b , c , d ) ,

(50) Γ ( β + 1 ) 2 ( d − c ) β [ J c + β F ( a , d ) G ( b , d ) + J c + β F ( b , d ) G ( a , d ) ] + Γ ( β + 1 ) 2 ( d − c ) β [ J d − β F ( a , c ) G ( b , c ) + J d − β F ( b , c ) G ( a , c ) ] ⊇ 1 2 − β ( β + 1 ) ( β + 2 ) L ( a , b , c , d ) + β ( β + 1 ) ( β + 2 ) N ( a , b , c , d ) ,

(51) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , c ) + J a + α F ( b , d ) G ( b , d ) ] + Γ ( α + 1 ) 2 ( b − a ) α [ J b − α F ( a , c ) G ( a , c ) + J b − α F ( a , d ) G ( a , d ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) K ( a , b , c , d ) + α ( α + 1 ) ( α + 2 ) L ( a , b , c , d ) ,

and

(52) Γ ( α + 1 ) 2 ( b − a ) α [ J a + α F ( b , c ) G ( b , d ) + J a + α F ( b , d ) G ( b , c ) ] + Γ ( α + 1 ) 2 ( b − a ) α [ J b − α F ( a , c ) G ( a , d ) + J b − α F ( a , d ) G ( a , c ) ] ⊇ 1 2 − α ( α + 1 ) ( α + 2 ) M ( a , b , c , d ) + α ( α + 1 ) ( α + 2 ) N ( a , b , c , d ) .

If we substitute (49)–(52) in (48), and divide the resulting inequality by 2, then we obtain the desired result (32). This completes the proof.□

Remark 6

If we choose α = β = 1 in Theorem 10, then we have

4 F a + b 2 , c + d 2 G a + b 2 , c + d 2 ⊇ 1 ( b − a ) ( d − c ) ∫ a b ∫ c d F ( t , s ) G ( t , s ) d s d t + 5 36 K ( a , b , c , d ) + 7 36 [ L ( a , b , c , d ) + M ( a , b , c , d ) ] + 2 9 N ( a , b , c , d ) ,

given by Zhao et al. in [41].

Corollary 2

If we choose G ( x , y ) = [ 1 , 1 ] in Theorem 10, then we have the following inequality:

4 f a + b 2 , c + d 2 ⊇ Γ ( α + 1 ) Γ ( β + 1 ) 4 ( b − a ) α ( d − c ) β [ J a + , c + α , β F ( b , d ) + J a + , d − α , β F ( b , c ) + J b − , c + α , β F ( a , d ) + J b − , d − α , β F ( a , c ) ] + 3 4 [ F ( a , c ) + F ( a , d ) + F ( b , c ) + F ( b , d ) ] .

Remark 7

If F ̲ ( t ) = F ¯ ( t ) in Theorem 10, then Theorem 10 reduces to the result of Budak and Sarikaya [43, Theorem 2.2].

5 Conclusion

In this paper, we presented ideas about interval-valued fractional integrals on co-ordinates. We have used newly described fractional integrals to prove some new Hermite-Hadamard-type inequalities for co-ordinated convex interval-valued functions. It is a fascinating and novel problem that the future researchers may find similar inequalities for various types of convexity in their study.

Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

  1. Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

  2. Conflict of interest: Authors state no conflict of interest.

  3. Data availability statement: Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.

References

[1] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. Search in Google Scholar

[2] J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992. Search in Google Scholar

[3] T.-H. Zhao, Z.-Y. He, and Y.-M. Chu, Sharp bounds for the weighted Hølder mean of the zero-balanced generalized complete elliptic integrals, Comput. Methods Funct. Theory 21 (2021), no. 3, 413–426. 10.1007/s40315-020-00352-7Search in Google Scholar

[4] M.-K. Wang, H.-H. Chu, and Y.-M. Chu, On the approximation of some special functions in Ramanujan’s generalized modular equation with signature 3, Ramanujan J. 56 (2021), no. 1, 1–22. 10.1007/s11139-021-00437-4Search in Google Scholar

[5] P. Agarwal, M. Kadakal, I. Iscan, and Y.-M. Chu, Better approaches for n-times differentiable convex functions, Mathematics 8 (2020), no. 6, 950. 10.3390/math8060950Search in Google Scholar

[6] M. A. Ali, H. Budak, Z. Zhang, and H. Yildrim, Some new Simpson’s type inequalities for co-ordinated convex functions in quantum calculus, Math. Meth. Appl. Sci. 44 (2021), 4515–4540. 10.22541/au.159440598.88087022Search in Google Scholar

[7] M. A. Ali, H. Budak, M. Abbas, and Y.-M. Chu, Quantum Hermite-Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives, Adv. Differ. Equ. 2021 (2021), 7. 10.1186/s13662-020-03163-1Search in Google Scholar

[8] M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza, and Y.-M. Chu, New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions, Adv. Differ. Equ. 2021 (2021), 64. 10.1186/s13662-021-03226-xSearch in Google Scholar

[9] M. A. Ali, Y.-M. Chu, H. Budak, A. Akkurt, and H. Yildrim, Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables, Adv. Differ. Equ. 2021 (2021), 25. 10.1186/s13662-020-03195-7Search in Google Scholar

[10] M. A. Ali, N. Alp, H. Budak, Y.-M. Chu, and Z. Zhang, On some new quantum midpoint type inequalities for twice quantum differentiable convex functions, Open Math. 19 (2021), 427–439. 10.1515/math-2021-0015Search in Google Scholar

[11] M. A. Ali, H. Budak, A. Akkurt, and Y.-M. Chu, Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus, Open Math. 19 (2021), 440–449. 10.1515/math-2021-0020Search in Google Scholar

[12] S.-S. Zhou, S. Rashid, M. A. Noor, K. I. Noor, F. Safdar, and Y.-M. Chu, New Hermite-Hadamard type inequalities for exponentially convex functions and applications, AIMS Math. 5 (2020), 6874–6901. 10.3934/math.2020441Search in Google Scholar

[13] H. Budak, S. Erden, and M. A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Meth. Appl. Sci. 44 (2020), 378–390. 10.1002/mma.6742Search in Google Scholar

[14] S.-B. Chen, S. Rashid, M. A. Noor, R. Ashraf, and Y.-M. Chu, A new approach on fractional calculus and probability density functions, AIMS Math. 5 (2020), 7041–7054.10.3934/math.2020451Search in Google Scholar

[15] H. Budak, M. A. Ali, and M. Tarhanaci, Some new quantum Hermite-Hadamard-like inequalities for coordinated convex functions, J. Optim. Theory Appl. 186 (2020), 899–910. 10.1007/s10957-020-01726-6Search in Google Scholar

[16] Y.-X. Li, M. A. Ali, H. Budak, M. Abbas, and Y.-M. Chu, A new generalization of some quantum integral inequalities for quantum differentiable convex functions, Adv. Differ. Equ. 2021 (2021), 225.10.1186/s13662-021-03382-0Search in Google Scholar

[17] F. Chen and S. Wu, Several complementary inequalities to inequalities of Hermite-Hadamard-type for s-convex functions, J. Nonlinear Sci. Appl. 9 (2016), 705–716. 10.22436/jnsa.009.02.32Search in Google Scholar

[18] S. B. Chen, H. Jahanshahi, O. Alhadji Abba, J. E. Solis-Perez, S. Bekiros, J. F. Gomez-Aguilar, et al., The effect of market confidence on a financial system from the perspective of fractional calculus: numerical investigation and circuit realization, Chaos Solitons Fractals. 140 (2020), 110223.10.1016/j.chaos.2020.110223Search in Google Scholar

[19] H.-H. Chu, T.-H. Zhao, and Y.-M. Chu, Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means, Math. Slovaca 70 (2020), no. 5, 1097–1112. 10.1515/ms-2017-0417Search in Google Scholar

[20] T.-H. Zhao, M.-K. Wang, and Y.-M. Chu, Concavity and bounds involving generalized elliptic integral of the first kind, J. Math. Inequal. 15 (2021), no. 2, 701–724. 10.7153/jmi-2021-15-50Search in Google Scholar

[21] T.-H. Zhao, M.-K. Wang, and Y.-M. Chu, Monotonicity and convexity involving generalized elliptic integral of the first kind, Rev. R. Acad. Cienc, Exactas, Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 2, 46.10.1007/s13398-020-00992-3Search in Google Scholar

[22] Y.-X. Li, A. Rauf, M. Naeem, M. A. Binyamin, and A. Aslam, Valency-based topological properties of linear hexagonal chain and hammer-like benzenoid, Complexity 2021 (2021), 9939469.10.1155/2021/9939469Search in Google Scholar

[23] M. Tomar, P. Agarwal, and J. Choi, Hermite-Hadamard-type inequalities for generalized convex functions on fractal sets style, Bol. Soc. Parana. Mat. 38 (2020), no. 1, 101–116. 10.5269/bspm.v38i1.32820Search in Google Scholar

[24] T.-H. Zhao, Z.-H. Shen, and Y.-M. Chu, Sharp power mean bounds for the lemniscate type means, Rev. R. Acad, Cienc, Exactas, Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 4, 175.10.1007/s13398-021-01117-0Search in Google Scholar

[25] S.-B. Chen, S. Rashid, Z. Hammouch, M. A. Noor, R. Ashraf, and Y.-M. Chu, Integral inequalities via Raina's fractional integrals operator with respect to a monotone function, Adv. Differ. Equ. 2020 (2020), 647.10.1186/s13662-020-03108-8Search in Google Scholar

[26] D. Zhao, M. A. Ali, A. Kashuri, and H. Budak, Generalized fractional integral inequalities of Hermite-Hadamard-type for harmonically convex functions, Adv. Differ. Equ. 2020 (2020), no. 1, 1–14. 10.1186/s13662-020-02589-xSearch in Google Scholar

[27] R. E. Moore, Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. Search in Google Scholar

[28] Y. Chalco-Cano, A. Flores-Franulic, and H. Roman-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math. 31 (2012), 457–472. 10.1109/IFSA-NAFIPS.2013.6608617Search in Google Scholar

[29] Y. Chalco-Cano, W. A. Lodwick, and W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput. 19 (2015), 3293–3300. 10.1007/s00500-014-1483-6Search in Google Scholar

[30] E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set-valued functions, Results Math. 32 (1997), 332–337. 10.1007/BF03322144Search in Google Scholar

[31] F. C. Mitroi, N. Kazimierz, and W. Szymon, Hermite-Hadamard inequalities for convex set-valued functions, Demonstr. Math. 46 (2013), no. 4, 655–662. 10.1515/dema-2013-0483Search in Google Scholar

[32] K. Nikodem, J. L. Sánchez, and L. Sánchez, Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Math. Aeterna 4 (2014), no. 8, 979–987. Search in Google Scholar

[33] R. E. Moore, R. B. Kearfott, and M. J. Cloud, Introduction to Interval Analysis, vol. 110, SIAM, Philadelphia, 2009. 10.1137/1.9780898717716Search in Google Scholar

[34] D. Zhao, T. An, G. Ye, and W. Liu, New Jensen and Hermite-Hadamard-type inequalities for h-convex interval-valued functions, J. Inequal. Appl. 2018 (2018), 302. 10.1186/s13660-018-1896-3Search in Google Scholar

[35] D. Zhao, G. Ye, W. Liu, and D. F. M. Torres, Some inequalities for interval-valued functions on time scales, Soft Comput. 23 (2019), 6005–601510.1007/s00500-018-3538-6Search in Google Scholar

[36] W. W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numér. Théor. Approx. 22 (1993), 39–51. Search in Google Scholar

[37] R. Osuna-Gómez, M. D. Jiménez-Gamero, Y. Chalco-Cano, and M. A. Rojas-Medar, Hadamard and Jensen inequalities for s-convex fuzzy processes, in: M. López-Díaz, M. Á. Gil, P. Grzegorzewski, O. Hryniewicz, and J. Lawry (eds), Soft Methodology and Random Information Systems, Springer, Berlin, 2004, pp. 645–652. 10.1007/978-3-540-44465-7_80Search in Google Scholar

[38] H. Budak, T. Tunç, and M. Z. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proc. Amer. Math. Soc. 148 (2020), 705–718. 10.1090/proc/14741Search in Google Scholar

[39] V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets and Systems 265 (2015), 63–85. 10.1016/j.fss.2014.04.005Search in Google Scholar

[40] D. Zhao, T. An, G. Ye, and W. Liu, Chebyshev type inequalities for interval-valued functions, Fuzzy Sets and Systems 396 (2020), 82–101. 10.1016/j.fss.2019.10.006Search in Google Scholar

[41] D. Zhao, M. A. Ali, G. Murtaza, and Z. Zhang, On the Hermite-Hadamard inequalities for interval-valued coordinated convex functions, Adv. Differ. Equ. 2020 (2020), 570. 10.1186/s13662-020-03028-7Search in Google Scholar

[42] M. Z. Sarikaya, On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integr. Transforms Special Funt. 25 (2014), no. 2, 134–147. 10.1080/10652469.2013.824436Search in Google Scholar

[43] H. Budak and M. Z. Sarikaya, Hermite-Hadamard-type inequalities for products of two co-ordinated convex mappings via fractional integrals, Int. J. Appl. Math. Stat. 58 (2019), no. 4, 11–30. Search in Google Scholar
Received: 2020-11-07
Revised: 2021-03-17
Accepted: 2021-07-05
Published Online: 2021-09-21

© 2021 Huseyin Budak et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0067
Keywords for this article
fractional integrals; Hermite-Hadamard inequality; interval-valued functions
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2021-0067/html
Scroll to top button