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On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel

  • Miguel Vivas-Cortez , Maria Bibi , Muhammad Muddassar and Sa’ud Al-Sa’di EMAIL logo
Published/Copyright: May 30, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

Keywords: generalized h̃1, h̃2-preinvex functions; local fractional integrals; generalized Hermite-Hadamard inequality; fractal sets; Mittag-Leffler kernel
MSC 2010: 26A33; 26A51; 90C23; 26D10; 26D15

1 Introduction

In the area of mathematical inequalities, convex functions are connected with a variety of assumptions. The most well-known inequality with its extensive geometrical structure is the Hermite-Hadamard inequality, presented by Jacques Hadamard in 1881 (see [1]). It is well understood that a function Q is a convex function on an interval J if and only if it meets the following inequality:

(1) Q s + r 2 ≤ 1 r − s ∫ s r Q ( y ) d y ≤ Q ( s ) + Q ( r ) 2 ,

for s , r ∈ J and s < r . This classical inequality has been studied broadly in terms of extensions, generalizations, and improvements via various generalizations of convex function (we refer to [2–8] and references therein for more details).

One of the famous extended convex functions is preinvex function. Throughout this work, we assume that D ⊆ R n is a non-empty set, Q : D → R n a continuous mapping, and η ( . , . ) : D × D → R n be a bi-function. Recall that a set D ⊆ R n is said to be invex with respect to the bi-function η ( . , . ) : D × D → R n if

s + ς η ( r , s ) ∈ D ,

where s , r ∈ D , and 0 ≤ ς ≤ 1 . If η ( r , s ) = s − r holds, the invex set D becomes convex (see [9, p. 1]).

Definition 1

[10, p. 30] Given an invex set D with respect to η ( . , . ) . Then, Q : D → R n is the preinvex function on D if

(2) Q ( s + ς η ( r , s ) ) ≤ ( 1 − ς ) Q ( s ) + ς Q ( r ) ,

where s , r ∈ D and 0 ≤ ς ≤ 1 . When − Q is preinvex, Q becomes preconcave.

Definition 2

[11, p. 2] Given two non-negative functions h ˜ 1 , h ˜ 2 : ( 0 , 1 ) ⊆ J → R . A function Q on the invex set D is said to be ( h ˜ 1 , h ˜ 2 ) -preinvex, if

(3) Q ( s + ς η ( r , s ) ) ≤ h ˜ 1 ( 1 − ς ) h ˜ 2 ( ς ) Q ( s ) + h ˜ 1 ( ς ) h ˜ 2 ( 1 − ς ) Q ( r ) ,

where s , r ∈ D and 0 ≤ ς ≤ 1 .

Note that the Jensen-type ( h ˜ 1 , h ˜ 2 ) -preinvex function is obtained if ς = 1 2 in (3), that is,

Q 2 s + η ( r , s ) 2 ≤ h ˜ 1 1 2 h ˜ 2 1 2 [ Q ( s ) + Q ( r ) ] .

Yang presented the theory of local fractional calculus in [12,13]. This theory has valid applications in communication engineering, control theory, physics, and random walk process (see [14–17]). In the context of ordinary and partial differential equations, local fractional calculus can be used to study the dynamics of systems that exhibit non-differentiable behavior at certain points in space as well as time (see [18–25]). On the other hand, different studies were conducted to extend the concept of convex functions on fractal sets to explore Hermite-Hadamard inequalities (we refer [26–33] for more details). Some important generalizations of convex functions in fractal sense include the results presented by Sun [34–40]. Ahmad et al. [41] studied functional inequalities for convex functions emphasizing novel integral operators of fractional order with exponential kernel. Sun in [42] defined two fractal integral operators with Mittag-Leffler kernel to study local fractional inequalities of Hermite-Hadamard-type via generalized h -convex functions involving these operators.

Definition 3

[42, p. 4989] Let Q ∈ L ( s , r ) . The fractional integrals I s ξ ( g ) and I r ξ ( g ) of order ξ ∈ ( 0 , 1 ) are given as:

(4) I s ξ ( x ) = 1 ξ ξ Γ ( 1 + ξ ) ∫ s g E ξ − 1 − ξ ξ ( g − k ) Q ( k ) d k , g > s ,

and

(5) I r ξ ( x ) = 1 ξ ξ Γ ( 1 + ξ ) ∫ g r E ξ − 1 − ξ ξ ( k − g ) Q ( k ) d k , g < r .

Theorem 1

[42, p. 4989] Let the function Q : [ s , r ] → R ξ be a positive function with 0 ≤ s < r and Q ( y ) ∈ I y ( ξ ) [ s , r ] . If Q is a generalized h-convex function on [ s , r ] , then

(6) [ 1 ξ − E ξ ( − ρ ) ξ ] ρ ξ h ξ 1 2 Q s + r 2 ≤ ξ ( r − s ) ξ [ I s + ( ξ ) Q ( r ) + I r − ( ξ ) Q ( s ) ] ≤ [ Q ( s ) + Q ( r ) ] 0 I 1 ( ξ ) E ξ ( − ρ ς ) ξ [ h ξ ( ς ) + h ξ ( 1 − ς ) ] ,

where ρ = 1 − ξ ξ ( r − s ) .

Sun in [43, Lemma 5] proved the following identity for generalized preinvex functions.

Lemma 1

Let D ⊆ R be an open invex subset with respect to the bi-function η : D × D → R , where s , r ∈ D , s < s + η ( r , s ) . Assume that Q : I → R ξ is a local fractional differentiable function with η ( r , s ) ≥ 0 , s , r ∈ D . If Q ( y ) ξ ∈ I y ( ξ ) [ s , s + η ( r , s ) ] , then the following equality holds:

Q ( s ) + Q ( s + η ( r , s ) ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] = η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ξ ( s + ( 1 − ς ) η ( r , s ) ) ( d ς ) ξ − 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ .

2 Preliminaries

We recall some properties for addition and multiplication operations on R ξ for 0 < ξ ≤ 1 . If d ξ , e ξ , f ξ ∈ R ξ , then

  • d ξ + e ξ ∈ R ξ , d ξ e ξ ∈ R ξ ,

  • d ξ + e ξ = d ξ + e ξ = ( d + e ) ξ = ( e + d ) ξ ,

  • d ξ + ( e ξ + f ξ ) = ( d + e ) ξ + f ξ ,

  • d ξ e ξ = e ξ d ξ = ( d e ) ξ = ( e d ) ξ ,

  • d ξ ( e ξ f ξ ) = ( d ξ e ξ ) f ξ ,

  • d ξ ( e ξ + f ξ ) = d ξ e ξ + d ξ f ξ ,

  • d ξ + 0 ξ = 0 ξ + d ξ = d ξ , and d ξ 1 ξ = 1 ξ d ξ = d ξ ,

  • If d ξ < e ξ , then d ξ + f ξ < e ξ + f ξ ,

  • If 0 ξ < d ξ , 0 ξ < e ξ , then 0 ξ < d ξ e ξ .

We now review the concepts of local fractional continuity, derivative, and integral on R ξ (see [12, chapter 2] and [13, chapter 1] for more extensive study).

Definition 4

A non-differentiable mapping Q : R → R ξ , y → Q ( y ) is said to be local fractional continuous at y 0 ; if for any ε > 0 , there exists δ > 0 such that

∣ Q ( y ) − Q ( y 0 ) ∣ < ε ξ

holds whenever ∣ y − y 0 ∣ < δ , with ε , δ ∈ R . If Q ( y ) is the local fractional continuous function on ( b , c ) , we denote Q ( y ) ∈ C ξ ( b , c ) .

Definition 5

The local fractional derivative of Q ( y ) of order ξ at y = y 0 is defined as follows:

Q ( ξ ) ( y 0 ) = d ξ Q ( y ) d ς ξ y = y 0 = lim y → y 0 Γ ( 1 + ξ ) ( Q ( y ) − Q ( y 0 ) ) ( y − y 0 ) ξ .

In this case, D ξ ( b , c ) is called ξ -local derivative set. If there exists Q ( ( K + 1 ) ξ ) ( y ) = D y ξ … D y ξ ︷ ( n + 1 ) times Q ( y ) for any y ∈ I ⊆ R , we denote Q ∈ D ( n + 1 ) ξ ( I ) , and n = 0 , 1 , 2 , … .

Definition 6

Let Q ( y ) ∈ C ξ [ b , c ] . The local fractional integral of Q ( y ) can be defined by:

I c ξ b Q ( y ) = 1 Γ ( 1 + ξ ) ∫ b c Q ( ς ) ( d ς ) ξ = 1 Γ ( 1 + ξ ) lim Δ ς → 0 ∑ j = 0 N − 1 Q ( ς j ) ( Δ ς j ) ξ ,

where b = ς 0 < ς 1 < … < ς N − 1 < ς N = c , [ ς j , ς j + 1 ] is a partition of [ b , c ] , Δ ς j = Δ ς j + 1 − Δ ς j , Δ ς = max { ς 0 , ς 1 … ς N − 1 } .

Note that I c ξ b Q ( y ) = 0 if b = c , and I b ξ c Q ( y ) = − I c ξ b Q ( y ) if b < c . We denote Q ( y ) ∈ I y ξ [ b , c ] if there exists I y ξ b Q ( y ) for any y ∈ [ b , c ] .

Mittag-Leffler function of fractal order ξ ( 0 < ξ ≤ 1 ) on Yang’s fractal sets can be defined by:

(7) E ξ ( y ξ ) = ∑ k = 0 ∞ y k ξ Γ ( 1 + k ξ ) y ∈ R .

Formulas for local fractional calculus of Mittag-Leffler function are presented as (see [12, chapter 2] and [13, chapter 1]).

Lemma 2

The local fractional derivative and local fractional integration of Mittag-Leffler function can be given as:

(8) d ξ E ξ ( k y ξ ) d y ξ = k E ξ ( k y ξ ) , k is a c o n s t a n t ,

(9) I c ξ b E ξ ( y ξ ) = E ξ ( c ξ ) − E ξ ( b ξ ) .

Lemma 3

  1. Let g ( y ) = f ( ξ ) ( y ) ∈ C ξ [ b , c ] . Then,

    I c ξ b g ( y ) = f ( c ) − f ( b ) .

  2. Let g ( y ) , f ( y ) ∈ D ξ [ b , c ] and g ( ξ ) ( y ) , f ( ξ ) ( y ) ∈ C ξ [ b , c ] . Then,

    I c ξ b g ( y ) f ( ξ ) ( y ) = g ( y ) f ( y ) ∣ b c − I c ξ b g ( ξ ) ( y ) f ( y ) .

Lemma 4

Let Q ( y ) ∈ C ξ [ b , c ] . The local fractional derivative and integral of an elementary function Q ( y ) = y s ξ is given by:

d ξ y s ξ d u ξ = Γ ( 1 + s ξ ) Γ ( 1 + ( s − 1 ) ξ ) y ( s − 1 ) ξ , 1 Γ ( ξ + 1 ) ∫ b c y s ξ ( d u ) ξ = Γ ( 1 + s ξ ) Γ ( 1 + ( s + 1 ) ξ ) ( c ( s + 1 ) ξ − b ( s + 1 ) ξ ) , s > 0 .

Lemma 5

Let Q ( y ) = 1 . Then, by property of mean value theorem for local fractional integrals, we have

I c ξ b 1 ξ = ( c − b ) ξ Γ ( 1 + ξ ) .

Lemma 6

(Generalized Hölder’s inequality) Let p , q > 1 with p − 1 + q − 1 = 1 . Let g ( y ) , f ( y ) ∈ C ξ [ b , c ] . Then,

(10) 1 Γ ( ξ + 1 ) ∫ b c ∣ g ( y ) f ( y ) ∣ ( d y ) ξ ≤ 1 Γ ( ξ + 1 ) ∫ b c ∣ g ( y ) ∣ p ( d y ) ξ 1 p 1 Γ ( ξ + 1 ) ∫ b c ∣ f ( y ) ∣ q ( d y ) ξ 1 q .

We recall the generalized beta function, which will be used throughout the work:

(11) B ξ ( y , x ) = 1 Γ ( 1 + ξ ) ∫ 0 1 ς ( y − 1 ) ξ ( 1 − ς ) ( x − 1 ) ξ ( d ς ) ξ , y > 0 , x > 0 .

3 Main results

We begin by introducing the definition of generalized ( h ˜ 1 , h ˜ 2 ) -preinvex with respect to a bi-function η .

Definition 7

Suppose that h ˜ 1 , h ˜ 2 : ( 0 , 1 ) ⊆ J → R are two non-negative functions with h ˜ 1 ξ ≢ 0 ξ , h ˜ 2 ξ ≢ 0 ξ , and ( 0 , 1 ) ⊆ J is any interval in R . Let D be an invex set with respect to η . A function Q : D → R ξ ( 0 < ξ ≤ 1 ) is said to be generalized ( h ˜ 1 , h ˜ 2 ) -preinvex with respect to η ; if for each s , r ∈ D and ς ∈ [ 0 , 1 ] , we have

(12) Q ( r + ς η ( s , r ) ) ≤ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) Q ( s ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) Q ( r ) .

Moreover, the function Q is generalized ( h ˜ 1 , h ˜ 2 ) -preconcave with respect to η if it satisfies the aforementioned reverse inequality.

Remark 1

If we assume ξ = 1 in (12), then a generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function becomes a classical ( h ˜ 1 , h ˜ 2 ) -preinvex function. If h ˜ 1 ξ ( ς ) = ς s ξ and h ˜ 2 ξ ( ν ) = ν s ξ with s ∈ [ 0 , 1 ] , then (12) becomes generalized s-preinvex function. Moreover, if h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) = 1 , then (12) becomes generalized Toader-like preinvex function (see [44, p. 79]).

For what follows, we set ρ = 1 − ξ η ( r , s ) .

Theorem 2

Let D ⊆ R be an open invex subset with respect to the bi-function η : D × D → R , where s , r ∈ D , and s < s + η ( r , s ) . Let Q : D → R ξ be a positive function with η ( r , s ) ≥ 0 and Q ( y ) ∈ I y ( ξ ) [ s , s + η ( r , s ) ] . If Q is generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] , then the following inequalities hold:

(13) [ 1 ξ − E ξ ( − ρ ) ξ ] ρ ξ h ˜ 1 ξ 1 2 h ˜ 2 ξ 1 2 Q 2 s + η ( r , s ) 2 ≤ ξ η ( s , r ) ξ [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ≤ [ Q ( s ) + Q ( r ) ] 0 I 1 ( ξ ) E ξ ( − ρ ς ) ξ [ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ] .

Proof

Since Q is a generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function, then

Q 2 w + η ( x , w ) 2 ≤ h ˜ 1 1 2 h ˜ 2 1 2 [ Q ( w ) + Q ( x ) ] .

Let w = s + ς η ( r , s ) , and x = s + ( 1 − ς ) η ( r , s ) . Then,

(14) 1 ξ h ˜ 1 ξ 1 2 h ˜ 2 ξ 1 2 Q 2 s + η ( r , s ) 2 ≤ Q ( s + ς η ( r , s ) ) + Q ( s + ( 1 − ς ) η ( r , s ) ) .

By multiplying both sides of the inequality in (14) by E ξ ( − ρ ς ) ξ , then using local fractional integration corresponding to ς over [ 0 , 1 ] of the resulting inequality, we obtain

(15) 1 ξ h ˜ 1 ξ 1 2 h ˜ 2 ξ 1 2 Q 2 s + η ( r , s ) 2 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ ( d ς ) ξ ≤ 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ( s + ς η ( r , s ) ) ( d ς ) ξ + 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ( s + ( 1 − ς ) η ( r , s ) ) ( d ς ) ξ .

By setting s + ς η ( r , s ) = u , and s + ( 1 − ς ) η ( r , s ) = v , we have

(16) 1 ξ ρ ξ h ˜ 1 ξ 1 2 h ˜ 2 ξ 1 2 Q 2 s + η ( r , s ) 2 ≤ 1 η ( r , s ) ξ 1 Γ ( 1 + ξ ) ∫ s s + η ( r , s ) E ξ − ρ s + η ( r , s ) − u η ( r , s ) ξ Q ( u ) ( d u ) ξ + 1 Γ ( 1 + ξ ) ∫ s s + η ( r , s ) E ξ − ρ v − s η ( r , s ) ξ Q ( v ) ( d v ) ξ ≤ 1 η ( s , s ) ξ 1 Γ ( 1 + ξ ) ∫ s s + η ( r , s ) E ξ − ρ − 1 − ξ ξ ξ ( s + η ( r , s ) − u ) ξ Q ( u ) ( d u ) ξ + 1 Γ ( 1 + ξ ) ∫ s s + η ( r , s ) E ξ − ρ − 1 − ξ ξ ξ ( v − s ) ξ Q ( v ) ( d v ) ξ = ξ η ( r , s ) ξ [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] .

Hence, we established the right side of the inequality in (13). Now to prove the left side of the inequality, note that since Q is generalized ( h ˜ 1 , h ˜ 2 ) -preinvex functions on [ s , s + η ( r , s ) ] , we have

Q ( s + ς η ( r , s ) ) ≤ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) Q ( r ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) Q ( s ) ,

and

Q ( s + ( 1 − ς ) η ( r , s ) ) ≤ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) Q ( r ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) Q ( s ) .

Furthermore, we have

(17) Q ( s + ς η ( r , s ) ) + Q ( s + ( 1 − ς ) η ( r , s ) ) ≤ [ Q ( s ) + Q ( r ) ] [ h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) + h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ] .

By multiplying both sides of the inequality in (17) by E ξ ( − ρ ς ) ξ , then using local fractional integration corresponding to ς over [ 0 , 1 ] of the resulting inequality, we obtain

1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ( s + ς η ( r , s ) ) ( d ς ) ξ + 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ( s + ( 1 − ς ) η ( r , s ) ) ( d ς ) ξ ≤ [ Q ( s ) + Q ( r ) ] 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ [ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ] + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ] ( d ς ) ξ .

From (15) and (16), we obtain

(18) ξ η ( s , r ) ξ [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ≤ [ Q ( s ) + Q ( r ) ] 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ [ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ] + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ] ( d ς ) ξ ≤ [ Q ( s ) + Q ( r ) ] 0 I 1 ( ξ ) E ξ ( − ρ ς ) ξ [ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ] .

Hence, the left side of (18) holds.□

Corollary 1

If h ˜ 1 ξ ( ς ) = 1 2 ξ and h ˜ 2 ξ ( ς ) = 1 2 ξ in (18), then we obtain

Q 2 s + η ( r , s ) 2 ≤ 2 ξ ( 1 − ξ ) ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ≤ [ Q ( s ) + Q ( r ) ] .

Corollary 2

Using the fact that

(19) lim ξ → 1 1 ξ − E ξ ( − ρ ) ξ ρ ξ = 1 ,

we obtain inequality for classical ( h ˜ 1 , h ˜ 2 ) -preinvex function

1 2 h ˜ 1 1 2 h ˜ 2 1 2 Q 2 s + η ( r , s ) 2 ≤ 1 η ( r , s ) ∫ s s + η ( r , s ) Q ( y ) d y ≤ Q ( s ) ∫ 0 1 h ˜ 1 ( 1 − ς ) h ˜ 2 ( ς ) d ς + Q ( r ) ∫ 0 1 h ˜ 1 ( ς ) h ˜ 2 ( 1 − ς ) d ς .

Theorem 3

Let D ⊆ R be an open invex subset with respect to the bi-function η : D × D → R , where s , r ∈ D , s < s + η ( r , s ) . Assume that the function Q : D → R ξ is a local fractional differentiable function with η ( r , s ) ≥ 0 and Q ( y ) ξ ∈ I y ( ξ ) [ s , s + η ( r , s ) ] . If ∣ Q ( ξ ) ∣ q is the generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] with p − 1 + q − 1 = 1 , p , q > 1 , then the following inequality holds:

(20) Q ( s ) + Q ( s + η ( r , s ) ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ≤ η ξ ( r , s ) [ 1 ξ − E ξ ( − ρ ) ξ ] 1 ξ − E ξ ( − p ρ ) ξ ( − p ρ ) ξ 1 p × [ ∣ Q ( ξ ) ( s ) ∣ q + ∣ Q ( ξ ) ( r ) ∣ q ] 1 Γ ( 1 + ξ ) ∫ 0 1 [ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ] ( d ς ) ξ 1 q .

Proof

Utilizing Lemma 1 and the generalized Hölder’s inequality (10), we have

Q ( s ) + Q ( s + η ( r , s ) ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ≤ η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ ∣ Q ξ ( s + ( 1 − ς ) η ( r , s ) ) ∣ ( d ς ) ξ + 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ ∣ Q ξ ( s + ς η ( r , s ) ) ∣ ( d ς ) ξ ≤ η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − p ρ ς ) ξ ( d ς ) ξ 1 p 1 Γ ( 1 + ξ ) ∫ 0 1 ∣ Q ξ ( s + ( 1 − ς ) η ( r , s ) ) ∣ q ( d ς ) ξ 1 q + 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − p ρ ς ) ξ ( d ς ) ξ 1 p 1 Γ ( 1 + ξ ) ∫ 0 1 ∣ Q ξ ( s + ς η ( r , s ) ) ∣ q ( d ς ) ξ 1 q .

On the other hand, since ∣ Q ( ξ ) ∣ q is the generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] , then

(21) 1 Γ ( 1 + ξ ) ∫ 0 1 ∣ Q ξ ( s + ς η ( r , s ) ) ∣ q ( d ς ) ξ ≤ 1 Γ ( 1 + ξ ) ∫ 0 1 h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ∣ Q ( s ) ∣ q + 1 Γ ( 1 + ξ ) ∫ 0 1 h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ∣ Q ( r ) ∣ q 1 Γ ( 1 + ξ ) ∫ 0 1 ∣ Q ξ ( s + ς η ( r , s ) ) ∣ q ( d ς ) ξ ≤ 1 Γ ( 1 + ξ ) ∫ 0 1 h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ∣ Q ( s ) ∣ q + 1 Γ ( 1 + ξ ) ∫ 0 1 h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ∣ Q ( r ) ∣ q .

and

(22) 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − p ρ ς ) ξ ( d ς ) ξ = 1 ξ − E ξ ( − p ρ ) ξ ( − p ρ ) ξ .

By substituting (21) to (22) in (7), we obtain the desired result.□

Corollary 3

Choosing h ˜ 1 ( ς ) = ς ξ and h ˜ 2 ( ς ) = ς ξ in Theorem 3, we obtain

Q ( s ) + Q ( s + η ( r , s ) ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ≤ 2 ξ η ξ ( r , s ) [ 1 ξ − E ξ ( − ρ ) ξ ] 1 ξ − E ξ ( − p ρ ) ξ ( − p ρ ) ξ ( [ ∣ Q ( ξ ) ( s ) ∣ q + ∣ Q ( ξ ) ( r ) ∣ q ] β ξ ( 2 , 2 ) ) 1 q .

Lemma 7

Let D ⊆ R be an open invex subset with respect to the bi-function η : D × D → R , where s , r ∈ D , s < s + η ( r , s ) . Suppose that Q : D → R ξ is a local fractional differentiable function with η ( r , s ) ≥ 0 and Q ( y ) ξ ∈ I y ( ξ ) [ s , s + η ( r , s ) ] . If Q is the generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] , then

(23) Q ( s ) + Q ( s + η ( r , s ) ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] = η ( r , s ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ − 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ( 1 − ς ) ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ .

Proof

By local fractional integration by parts, and letting u = s + ς η ( r , s ) , we have

(24) 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ = 1 η ( r , s ) ξ E ξ ( − ρ ς ) ξ Q ( s + ς η ( r , s ) ) ( d ς ) ∣ 0 1 − 1 Γ ( 1 + ξ ) ∫ 0 1 Q ξ ( s + ς η ( r , s ) ) ( E ξ ( − ρ ς ) ξ ) ξ ( d ς ) ξ = 1 η ( r , s ) ξ E ξ ( − ρ ς ) ξ Q ( s + ς η ( r , s ) ) ( d ς ) ∣ 0 1 − 1 Γ ( 1 + ξ ) ∫ 0 1 Q ξ ( s + ς η ( r , s ) ) ( E ξ ( − ρ ς ) ξ ) ξ ( d ς ) ξ = 1 η ( r , s ) ξ E ξ ( − ρ ) ξ Q ( s + η ( r , s ) ) − Q ( s ) − ( − ρ ) ξ Γ ( 1 + ξ ) ∫ 0 1 Q ξ ( s + ς η ( r , s ) ) E ξ ( − ρ ς ) ξ ( d ς ) ξ = E ξ ( − ρ ) ξ Q ( s + η ( r , s ) ) − Q ( s ) η ( r , s ) ξ − ρ ξ ξ ξ 1 η ( r , s ) 2 ξ 1 ξ Γ ( 1 + ξ ) ∫ s s + η ( r , s ) E ξ − ρ s + η ( r , s ) − u η ( r , s ) ξ Q ( u ) ( d u ) ξ = E ξ ( − ρ ) ξ Q ( s + η ( r , s ) ) − Q ( s ) η ( r , s ) ξ − 1 − ξ η ( r , s ) ξ I s + ( ξ ) Q ( s + η ( r , s ) ) .

Similarly, we obtain

(25) 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ( 1 − ς ) ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ = E ξ ( − ρ ) ξ Q ( s ) − Q ( s + η ( r , s ) ) η ξ ( r , s ) − 1 − ξ η ( r , s ) ξ I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) .

Subtracting (25) from (24), we obtain

(26) 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ς ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ − 1 Γ ( 1 + ξ ) ∫ 0 1 E ξ ( − ρ ( 1 − ς ) ) ξ Q ξ ( s + ς η ( r , s ) ) ( d ς ) ξ = [ 1 ξ − E ξ ( − ρ ) ξ ] [ Q ( s ) + Q ( s + η ( r , s ) ) ] η ξ ( r , s ) − 1 − ξ η ( r , s ) ξ [ I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ] ,

which completes the proof.□

Theorem 4

Let D ⊆ R be an open invex subset with respect to the bi-function η : D × D → R , where s , r ∈ D , s < s + η ( r , s ) . Assume that Q : D → R ξ is a local fractional differentiable function with η ( r , s ) ≥ 0 and Q ξ ( y ) ∈ I y ( ξ ) [ s , s + η ( r , s ) ] . If ∣ Q ( ξ ) ∣ is a the generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] , then the following inequality holds:

(27) Q ( s ) + Q ( r ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] ( I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ) ≤ η ξ ( r , s ) ( ∣ Q ξ ( s ) ∣ + ∣ Q ξ ( r ) ∣ ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) [ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ] ( d ς ) ξ .

Proof

Since ∣ Q ( ξ ) ∣ is a generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] and ( h ˜ 1 , h ˜ 2 ) is a non-negative function, then by utilizing Lemma 7, we can establish

Q ( s ) + Q ( r ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] ( I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ) ≤ η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 ∣ E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ∣ ∣ Q ( ξ ) ( s + ς η ( r , s ) ) ∣ ( d ς ) ξ ≤ η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) ∣ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ∣ Q ( ξ ) ( r ) ∣ + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ∣ Q ( ξ ) ( s ) ∣ ∣ ( d ς ) ξ + 1 Γ ( 1 + ξ ) ∫ 1 2 1 ( E ξ ( − ρ ( 1 − ς ) ) ξ − E ξ ( − ρ ς ) ξ ) ∣ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ∣ Q ( ξ ) ( r ) ∣ + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ∣ Q ( ξ ) ( s ) ∣ ∣ ( d ς ) ξ ] = η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) ∣ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ∣ Q ( ξ ) ( r ) ∣ + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ∣ Q ( ξ ) ( s ) ∣ ∣ ( d ς ) ξ + 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) × ∣ h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) ∣ Q ( ξ ) ( r ) ∣ + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ∣ Q ( ξ ) ( s ) ∣ ∣ ( d ς ) ξ ] = η ξ ( r , s ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) × ( h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ) ( ∣ Q ( ξ ) ( r ) ∣ + ∣ Q ( ξ ) ( s ) ∣ ) ( d ς ) ξ = η ξ ( r , s ) ( ∣ Q ( ξ ) ( r ) ∣ + ∣ Q ( ξ ) ( s ) ∣ ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] × 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) ( h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ) ( d ς ) ξ ,

which completes the proof.□

Corollary 4

(Dragomir-Agarwal-type inequality) Suppose that h ˜ 1 ξ ( ς ) = 1 2 ξ and h ˜ 2 ξ ( ς ) = 1 2 ξ in (27), then we obtain

(28) Q ( s ) + Q ( r ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] ( I s + ( ξ ) Q ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) Q ( s ) ) ≤ 1 2 ξ η ξ ( r , s ) ∣ ( Q ( ξ ) ( s ) ∣ + ∣ Q ( ξ ) ( r ) ∣ ) 2 ξ ρ ξ tanh ξ ρ 4 ξ .

Proof

If h ˜ 1 ξ ( ς ) = 1 2 ξ and h ˜ 2 ξ ( ς ) = 1 2 ξ in (27), then

(29) 1 Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) ( h ˜ 1 ξ ( 1 − ς ) h ˜ 2 ξ ( ς ) + h ˜ 1 ξ ( ς ) h ˜ 2 ξ ( 1 − ς ) ) ( d ς ) ξ = 1 2 ξ Γ ( 1 + ξ ) ∫ 0 1 2 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) ( d ς ) ξ = [ 1 ξ − 2 ξ E ξ ( − ρ 2 ) ξ + E ξ ( − ρ ) ξ ] 2 ξ ρ ξ = 1 ξ − E ξ ( − ρ 2 ) ξ 2 2 ξ ρ ξ .

Substituting (29) into (28), the intended result yields after simplification.□

Corollary 5

For ξ → 1 , we obtain

(30) lim ξ → 1 ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] = 1 2 η ( r , s ) ,

and

(31) lim ξ → 1 ( E ξ ( − ρ ς ) ξ − E ξ ( − ρ ( 1 − ς ) ) ξ ) 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] = 1 − 2 ς 2 .

Letting ξ → 1 in Theorem 4, we have inequality for classical generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function,

Q ( s ) + Q ( r ) 2 − 1 η ( r , s ) ∫ s s + η ( r , s ) Q ( y ) d y ≤ η ( r , s ) ( ∣ Q ′ ( s ) ∣ + ∣ Q ′ ( r ) ∣ ) 2 ∫ 0 1 2 ( 1 − 2 ς ) [ h ˜ 1 ( 1 − ς ) h ˜ 2 ( ς ) + h ˜ 1 ( ς ) h ˜ 2 ( 1 − ς ) ] ( d ς ) .

4 Examples

Example 1

For 0 ≤ s < s + η ( r , s ) , the following inequality holds:

(32) ( 2 s + η ( r , s ) ) 3 4 ≤ 4 s 3 + 6 s 2 η ( s , s ) + 4 s η 2 ( s , s ) + η 3 ( s , s ) 4 ≤ s 3 + r 3 6 .

Proof

Let Q ( y ) = y 3 , y ∈ ( 0 , ∞ ) . Then, Q ( y ) is a classical generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function for h ˜ 1 ( ς ) = ς and h ˜ 2 ( ς ) = ς ; then, using the fact given in Corollary 2, we obtain

2 2 s + η ( r , s ) 2 3 ≤ 1 η ( s , s ) ∫ s s + η ( s , s ) y 3 d y ≤ s 3 ∫ 0 1 ς ( 1 − ς ) ( d ς ) + r 3 ∫ 0 1 ς ( 1 − ς ) ( d ς ) .

After simplifying the aforementioned inequality, we obtain (32).□

Example 2

For 0 ≤ s < s + η ( r , s ) , the following inequality holds:

(33) s 3 + r 3 2 − 4 s 3 + 6 s 2 η ( s , s ) + 4 s η 2 ( s , s ) + η 3 ( s , s ) 4 ≤ 3 η ( r , s ) ( s 2 + r 2 ) 40 .

Proof

Letting Q ( y ) = y 3 , y ∈ ( 0 , ∞ ) . Since Q ( y ) is a classical generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function for h ˜ 1 ( ς ) = ( ς + 1 ) 2 and h ˜ 2 ( ς ) = ( ς + 1 ) 2 . Hence, using the fact given in Corollary 5, we obtain (33).□

5 Application to random variables

Assume that X is a continuous random variable and Q : D → R ξ is a generalized probability density function. Let D ⊆ R be an open invex subset with respect to η : D × D → R , where s , r ∈ D , s < s + η ( r , s ) . The generalized τ t h central moment about an arbitrary point ζ ∈ R of X , τ ≥ 0 is given by:

(34) M ξ τ ( ζ ) = 1 Γ ( 1 + ξ ) ∫ s r ( y − ζ ) τ ξ g ( y ) d ( y ) ξ , τ = 1 , 2 , 3 , … .

Moreover, we have

( M ξ τ ( ζ ) ) ξ = − Γ ( 1 + τ ξ ) Γ ( 1 + ξ ) Γ ( 1 + ( τ − 1 ) ξ ) 1 Γ ( 1 + ξ ) ∫ s r ( y − ζ ) ( τ − 1 ) ξ g ( y ) d ( y ) ξ

and

( M ξ τ ( ζ ) ) ξ = − Γ ( 1 + τ ξ ) Γ ( 1 + ξ ) Γ ( 1 + ( τ − 1 ) ξ ) M ξ τ − 1 ( ζ ) ξ .

Proposition 1

Let D ⊆ R be an open invex subset with respect to the bi-function η : D × D → R , where s , r ∈ D , s < s + η ( r , s ) . Assume the hypothesis of Corollary 3. Let Q ( ξ ) = M ξ τ ( ζ ) . If ∣ ( M ξ τ ( ζ ) ) ξ ∣ q is a generalized ( h ˜ 1 , h ˜ 2 ) -preinvex function on [ s , s + η ( r , s ) ] , then for p − 1 + q − 1 = 1 , p , q > 1 , the following inequality involving generalized moment holds,

M ξ τ ( s ) + M ξ τ ( s + η ( r , s ) ) 2 ξ − ( 1 − ξ ) ξ 2 ξ [ 1 ξ − E ξ ( − ρ ) ξ ] [ I s + ( ξ ) M ξ τ ( s + η ( r , s ) ) + I ( s + η ( r , s ) ) − ( ξ ) M ξ τ ( s ) ] ≤ 2 ξ η ξ ( r , s ) [ 1 ξ − E ξ ( − ρ ) ξ ] Γ ( 1 + τ ξ ) Γ ( 1 + ξ ) Γ ( 1 + ( τ − 1 ) ξ ) 1 ξ − E ξ ( − p ρ ) ξ ( − p ρ ) ξ 1 p ( [ ∣ M ξ τ ( ξ ) ( s ) ∣ q + ∣ M ξ τ ( ξ ) ( r ) ∣ q ] β ξ ( 2 , 2 ) ) 1 q .

6 Conclusion

Utilizing the notion of integral operators of fractional order with Mittag-Leffler kernel on Yang’s fractal sets defined by [42], this work intended to acquire some new identities of generalized local fractional integral Hermite-Hadamard-type inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvex functions. Our results give definite estimations for the variation between the left part and middle part together with the middle part and right part in concerned inequality. Some novel generalized special cases manifested the imposing execution of local fractional integration. The derived results have been illustrated by two examples to check the accuracy of the obtained results. As special application, we constructed an integral inequality corresponding to central moments of continuous random variables.

Acknowledgement

The authors would like to thank the anonymous referees for their comments and suggestions, which helped greatly improve the manuscript.

  1. Funding information: This work received financial support from Pontificia Universidad Católica del Ecuador.

  2. Author contributions: All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2022-10-06
Revised: 2023-01-07
Accepted: 2023-02-20
Published Online: 2023-05-30

© 2023 the author(s), published by De Gruyter

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

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  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
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  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
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https://doi.org/10.1515/dema-2022-0216
Keywords for this article
generalized h̃1, h̃2-preinvex functions; local fractional integrals; generalized Hermite-Hadamard inequality; fractal sets; Mittag-Leffler kernel
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Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
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  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
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  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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