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Article

Pre-Invexity and Fuzzy Fractional Integral Inequalities via Fuzzy Up and Down Relation

by
Muhammad Bilal Khan
1,*,
Jorge E. Macías-Díaz
2,3,*,
Saeid Jafari
4,
Abdulwadoud A. Maash
5 and
Mohamed S. Soliman
5
1
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
2
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Ciudad Universitaria, Avenida Universidad 940, Aguascalientes 20131, Mexico
3
Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia
4
College of Vestsjaelland South & Mathematical and Physical Science Foundation, 4200 Slagelse, Denmark
5
Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
*
Authors to whom correspondence should be addressed.
Symmetry 2023, 15(4), 862; https://doi.org/10.3390/sym15040862
Submission received: 10 March 2023 / Revised: 24 March 2023 / Accepted: 26 March 2023 / Published: 4 April 2023
(This article belongs to the Section Mathematics)

Abstract

:
The symmetric function class interacts heavily with other types of functions. One of these is the pre-invex function class, which is strongly related to symmetry theory. This paper proposes a novel fuzzy fractional extension of the Hermite-Hadamard, Hermite-Hadamard-Fejér, and Pachpatte type inequalities for up and down pre-invex fuzzy-number-valued mappings. Using fuzzy fractional operators, several generalizations have been developed, where well-known results fit as particular cases. Additionally, some non-trivial examples are included to support the discussion and the applicability of the key findings. The approach appears trustworthy and effective for dealing with various nonlinear problems in science and engineering. The findings are general and may constitute contributions to complex waveform theory.

1. Introduction

In the literature on mathematical inequalities, the Hermite–Hadamard (HH) inequality emerges as an important, and frequently utilized, result [1,2,3,4,5,6,7,8,9,10]. Other classical and significant inequalities include the Beckenbach–Dresher, Gagliardo–Nirenberg, Hardy, Hölder, Ky Fan, Levinson, Lynger, Minkowski, Olsen, Opial, Ostrowski, Young, arithmetic–geometric, and others [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].
For a convex mapping Υ : K R on convex set K , the HH inequality is written as
Υ ρ + μ 2 1 μ ρ ρ μ Υ ϰ d ϰ Υ ρ + Υ μ 2 ,
For all  ρ , μ K , with ρ μ . If Υ is concave, then (1) is reversed.
The HH inequality (1), as well as some generalizations, converses, and modifications [26,27,28,29,30,31,32,33,34,35,36,37,38,39], plays a significant role in many scientific disciplines, including statistics, electrical engineering, economics and finance, information sciences, and coding theory.
The notion of convexity has been used in different areas of technology and science. Convex functions and convex sets have been generalized in the scope of numerous fields due to their robustness. The convexity of a function has been demonstrated to exist only when it fulfills an integral inequality (1).
As novel fractional differential and integral operators with exponential kernels emerged as useful tools in several subjects, it became possible to provide novel classes of functional variants for harmonically convex functions, as well as generalizations in the scope of convexity theory.
An important generalization of the HH inequality is the HH–Fejér inequality [40]. Let us consider Υ : K R a convex mapping on a convex set K , and ρ , μ K with ρ μ . Then, we have
Υ ρ + μ 2 1 ρ μ C ϰ d ϰ ρ μ Υ ϰ C ϰ d ϰ Υ ρ + Υ μ 2 .
If C ϰ = 1 , then we obtain (1) from (2). For a concave mapping, (2) is reversed. Different inequalities can be derived using distinct symmetric convex mappings, C ϰ .
Classical calculus underwent enormous development in the last few decades. As an extension of the classical differentiation and integration to non-integer orders, fractional calculus has increased relevance. Indeed, fractional operators can better depict real-world phenomena involving memory and hereditary properties. For instance, Baleanu et al. [41], Miller and Ross [42], and Kilbas et al. [43] discussed numerous applications and extensive methodologies of fractional calculus. The authors of [44,45,46,47,48,49,50,51,52,53,54] provide an overview of many fractional operators.
It has been shown that fractional differential equations capture complex systems’ dynamics more precisely than integer-order ones. Several applications can be found in cosmology, environmental sciences, medicine, biology, materials, image and signal processing, and many other fields.
Fractional integral inequalities are used in the scope of district subjects (please see [55,56,57,58,59,60,61,62,63,64,65,66] and references therein). The calculus of variations in the area of applied sciences has been widely addressed and has become an appealing research-oriented field to address the existence and uniqueness of solutions to fractional equations. For instance, Adil Khan et al. [67] derived the HH inequality for s-convex functions, while Khan et al. [68] considered weighted generalizations of the HH inequality for interval-valued exponential trigonometric functions. For more information, see [69,70,71,72,73,74,75,76,77,78,79,80] and the references therein.
Khan et al. extended the idea of convex interval-valued mappings (I·V·Ms) and convex fuzzy-number-valued mappings ( F N V M s) by using fuzzy-order relations, such that convex F N V M s include (h1, h2)-convex F N V M s [81] and harmonic convex F N V M s [82]. Moreover, in order to illustrate inequalities of HH, HH–Fejér, and Pachpatte types, they adopted h-pre-invex F N V M s [83], (h1, h2)-pre-invex F N V M s [84], and higher-order pre-invex F N V M s [85]. In recent research, Khan et al. [86] proposed new forms of HH and HH–Fejér inequalities by using the concept of fuzzy Riemann–Liouville (RL) fractional integrals via up and down (UD) pre-invex F N V M s. For up-to-date advancements in fuzzy-interval-valued analysis of integral inequalities, please see [87,88,89,90,91,92,93,94,95,96,97,98] and the references therein.
Based on the aforementioned ideas, in this paper, we establish HH, HH–Fejér, and Pachpatte type integral inequalities for UD pre-invex F N V M s using the fuzzy fractional integral operator for inferable functions. Additionally, we show complete agreement between the suggested approach and existing results. The new approach appears reliable and effective for dealing with various nonlinear problems in the scope of science and engineering. The findings are general and may constitute contributions to complex waveform theory.

2. Preliminary Concepts and Definitions

We recall a few definitions that will be relevant in what follows.
Let us consider that X o is the space of all closed and bounded intervals of R , and that M X o is given by
M = M * , M * = ϰ R | M * ϰ M * , M * , M * R .
If M * = M * , then we say that M is degenerate. In the follow-up, all intervals are considered non-degenerate. If M * 0 , then we say that M is positive. We denote by X o + = M * , M * : M * , M * X o   a n d   M * 0 the set of all positive intervals.
Let u R and u M be given by
u M = u M * , u M *   if   u > 0 , 0     if   u = 0 , u M * , u M *   if   u < 0 .
We consider the Minkowski sum, M + W , product, M × W , and difference, W M , for M , W X o , as
W * , W * + M * , M * = W * + M * , W * + M * ,
W * , W * × M * , M * = m i n W * M * , W * M * , W * M * , W * M * , m a x W * M * , W * M * , W * M * , W * M *
W * , W * M * , M * = W * M * , W * M * .
Remark 1.
(i) 
For a given  W * , W * , M * , M * X o ,  the relation I , defined on X o  by
M * , M * I W * , W *   i f   a n d   o n l y   i f   M * W * , W * M *
for all  W * , W * , M * , M * X o ,  is considered a partial interval inclusion relation. Moreover,  M * , M * I W * , W *  coincides with M * , M * W * , W *  on X o .  The relation I  is of UD order (see [96]).
(ii) 
For a given  W * , W * , M * , M * X o ,  the relation I , defined on X o  by W * , W * I M * , M *  if and only if W * M * , W * M *  or W * M * , W * < M * , is a partial interval order relation. In addition, we have that W * , W * I M * , M *  coincides with W * , W * M * , M *  on X o .  The relation I  is of left and right (LR) type [95,96].
Given the intervals  W * , W * , M * , M * X o ,  we consider their Hausdorff–Pompeiu distance as 
d H W * , W * , M * , M * = m a x W * M * , W * M * .
We have that X o , d H is a complete metric space [89,93,94].
Definition 1
([88]). A fuzzy subset  L of R is a mapping  M ~ : R [ 0,1 ] , which is the denoted membership mapping of  L . We adopt the symbol  to represent the set of all fuzzy subsets of  R .
Let us consider M ~ . If the following properties hold, then we say that M ~ is a fuzzy number:
(1)
M ~ is normal if there exists ϰ R and M ~ ϰ = 1 ;
(2)
M ~ is upper semi-continuous on R if for ϰ R there exist ε > 0 and δ > 0 , yielding M ~ ϰ M ~ y < ε for all y R with ϰ y < δ ;
(3)
M ~ is fuzzy convex, meaning that M ~ 1 u ϰ + u y m i n M ~ ϰ , M ~ y , for all ϰ , y R , and u [ 0,1 ] ;
(4)
M ~ is compactly supported, which means that c l ϰ R | M ~ ϰ > 0 is compact.
The symbol o will be adopted to designate the set of all fuzzy numbers of R .
Definition 2
([88,89]). For M ~ o , the b -level, or b -cut, sets of M ~  are M ~ b = ϰ R | M ~ ϰ > b  for all b [ 0 , 1 ] , and M ~ 0 = ϰ R | M ~ ϰ > 0 .
Proposition 1
([90]). Let M ~ , W ~ o . The relation F , defined on o  by
M ~ F W ~   w h e n   a n d   o n l y   w h e n   M ~ b I W ~ b ,   f o r   e v e r y   b [ 0 , 1 ] ,
is an LR order relation.
Proposition 2
([86]). Let M ~ , W ~ o . The relation  F , defined on  o by
M ~ F W ~   when   and   only   when   M ~ b I W ~ b ,   for   every   b [ 0 , 1 ] ,
is a UD order relation.
If M ~ , W ~ E I and u R , then, for every b 0 , 1 , the arithmetic operations, addition “ M ~ W ~ ”, multiplication “ M ~ W ~ ”, and multiplication by scalar “ u M ~ ” can be characterized level-wise, respectively, by
M ~ W ~ b = M ~ b + W ~ b ,
M ~ W ~ b = M ~ b × W ~ b ,
u M ~ b = u . M ~ b
These operations follow directly from Equations (4)–(6), respectively.
Theorem 1
([89]). For M ~ , W ~ o , the supremum metric
d M ~ , W ~ = sup 0 b 1 d H M ~ b , W ~ b
is a complete metric space, where  H stands for the Hausdorff metric on a space of intervals.
Theorem 2
([89,90]). If Υ : [ ρ , μ ] R X o  is an I-V·M satisfying Υ ϰ = Υ * ϰ , Υ * ϰ , then Υ  is Aumann integrable (IA-integrable) over [ ρ , μ ]  when and only when Υ * ϰ  and Υ * ϰ  are integrable over ρ , μ , meaning
I A ρ μ Υ ϰ d ϰ = ρ μ Υ * ϰ d ϰ , ρ μ Υ * ϰ d ϰ .
Definition 3
([95]). Let Υ ~ : I R o  be an F N V M . The family of I-V·Ms, for every b 0 , 1 , is Υ b : I R X o  satisfying Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b  for every ϰ I .  For every b 0 , 1 ,  the lower and upper mappings of Υ b  are the end point real-valued mappings Υ * · , b , Υ * · , b : I R .
Definition 4
([95]). Let Υ ~ : I R o  be an F N V M . Then, Υ ~ ϰ  is continuous at ϰ I ,  if for every b 0 , 1 , Υ b ϰ  is continuous when and only when Υ * ϰ , b  and Υ * ϰ , b  are continuous at ϰ I .
Definition 5
([89]). Let Υ ~ : [ ρ , μ ] R o  be an F N V M . The fuzzy Aumann integral ( F A -integral) of Υ ~  over ρ , μ  is
F A ρ μ Υ ~ ϰ d ϰ b = I A ρ μ Υ b ϰ d ϰ = ρ μ Υ ϰ , b d ϰ : Υ ϰ , b S Υ b ,
where S Υ b = Υ . , b R : Υ . , b   i s   i n t e g r a b l e ,   a n d   Υ ϰ , b Υ b ϰ for every b 0 , 1 .  Moreover, Υ ~  is F A -integrable over [ ρ , μ ]  if F A ρ μ Υ ~ ϰ d ϰ o .
Theorem 3
([90]). Let Υ ~ : [ ρ , μ ] R o  be an F N V M  whose b -levels define the family of I-V·Ms Υ b : [ ρ , μ ] R X o  satisfying Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b  for every ϰ [ ρ , μ ]  and b 0 , 1 .   Υ ~ is F A -integrable over [ ρ , μ ]  when and only when Υ * ϰ , b  and Υ * ϰ , b  are integrable over [ ρ , μ ] . Moreover, if Υ ~  is F A -integrable over ρ , μ , we have
F A ρ μ Υ ~ ϰ d ϰ b = ρ μ Υ * ϰ , b d ϰ , ρ μ Υ * ϰ , b d ϰ = I A ρ μ Υ b ϰ d ϰ
for every  b 0 , 1 .
Definition 6
([32]). Let β > 0 and L ρ , μ , o  be the collection of all Lebesgue measurable fuzzy-number-valued mappings on [ ρ , μ ] . Then, the fuzzy left and right RL fractional integrals of order β > 0  of Υ L ρ , μ , o are
I ρ + β Υ ~ ϰ = 1 Γ ( β ) ρ ϰ ϰ m β 1 Υ ~ m   d m ,     ϰ > ρ ,
and
I μ β Υ ~ ϰ = 1 Γ ( β ) ϰ μ m ϰ β 1 Υ ~ m   d m ,     ( ϰ < μ )
respectively, where  Γ ϰ = 0 m ϰ 1 e m   d m is the Euler gamma function. The fuzzy left and right RL fractional integrals ϰ based on left and right end point mappings are
I ρ + β Υ ~ ϰ b = 1 Γ β ρ ϰ ϰ m β 1 Υ b m   d m = 1 Γ ( β ) ρ ϰ ϰ m β 1 Υ * m , b , Υ * m , b   d m , ϰ > ρ ,
where
I ρ + β Υ * ϰ , b = 1 Γ ( β ) ρ ϰ ϰ m β 1 Υ * m , b   d m ,     ϰ > ρ ,
and
I ρ + β Υ * ϰ , b = 1 Γ ( β ) ρ ϰ ϰ m β 1 Υ * m , b   d m ,     ϰ > ρ .
The RL fractional integral Υ ~ of ϰ based on left and right end point mappings can be defined in a similar way.
An interval-valued mapping Υ : I = ρ , μ X o is a convex interval-valued mapping if
Υ u ϰ + 1 u y u Υ ϰ + ( 1 u ) Υ y ,
For all ϰ , y ρ , μ , u 0 , 1 , where X o is the collection of all real-valued intervals. If (24) is reversed, then Υ is said to be concave (see [91]).
Definition 7
([87]). The F N V M Υ ~ : ρ , μ o is convex on ρ , μ  if
Υ ~ u ϰ + 1 u y F u Υ ~ ϰ ( 1 u ) Υ ~ y ,
for all  ϰ , y ρ , μ , u 0 , 1 , where  Υ ~ ϰ F 0 ~ for all  ϰ ρ , μ . If (25) is reversed, then  Υ ~ is said to be concave. Moreover,  Υ ~ is named affine if and only if it is convex and concave.
Definition 8
([96]). The F N V M   Υ ~ : ρ , μ o is UD convex on ρ , μ  if
Υ ~ u ϰ + 1 u y F u Υ ~ ϰ ( 1 u ) Υ ~ y ,
for all  ϰ , y ρ , μ , u 0 , 1 , where  Υ ~ ϰ F 0 ~ for all  ϰ ρ , μ . If (26) is reversed then,  Υ ~ is UD concave. Moreover,  Υ ~ is named UD affine if and only if it is UD convex and concave.
Definition 9
([44]). Let K  be an invex set. The F N V M   Υ ~ : K o is UD pre-invex on K  with respect to φ  if
Υ ~ ϰ + 1 u φ y , ϰ F u Υ ~ ϰ 1 u Υ ~ y ,
for all  ϰ , y K , u 0 , 1 , where  Υ ~ ϰ F 0 ~ ,   φ : K × K R . The mapping  Υ ~ is UD pre-incave on  K with respect to  φ if (27) is reversed.
Theorem 4
([4]). Let Υ ~ : [ ρ , μ ] o  be an F N V M  whose b -levels describe the family of I-V·Ms Υ b : [ ρ , μ ] X o + X o  as
Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b ,
for all  ϰ [ ρ , μ ] and all  b 0 , 1 . The  Υ ~ is UD pre-invex on  [ ρ , μ ] if and only if for all  b 0 , 1 ,   Υ * ϰ , b is pre-invex and  Υ * ϰ , b is pre-incave.
Definition 10
([4]). Let Υ ~ : [ ρ , μ ] o  be an F N V M  whose b -levels describe the family of I-V·Ms Υ b : [ ρ , μ ] C o + C o  as
Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b ,
for all  ϰ [ ρ , μ ] and all  b 0 , 1 . Then,  Υ ~ is lower UDpre-invex (pre-incave) on  [ ρ , μ ] if and only if for all  b 0 , 1 ,
Υ * ϰ + 1 u φ y , ϰ , b u Υ * ϰ , b + 1 u Υ * y , b ,
and
Υ * ϰ + 1 u φ y , ϰ , b = u Υ * ϰ , b + 1 u Υ * y , b .
Definition 11
([4]). Let Υ ~ : [ ρ , μ ] o  be an F N V M  whose b -levels describe the family of I-V·Ms Υ b : [ ρ , μ ] C o + C o  as
Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b ,
for all  ϰ [ ρ , μ ] and all  b 0 , 1 . Then,  Υ ~ is upper UD pre-invex (pre-incave) on  [ ρ , μ ] if and only if for all  b 0 , 1 ,
Υ * ϰ + 1 u φ y , ϰ , b = u Υ * ϰ , b + 1 u Υ * y , b ,
and
Υ * ϰ + 1 u φ y , ϰ , b u Υ * ϰ , b + 1 u Υ * y , b .
Remark 2
([4]). Both concepts UD pre-invex F N V M  and pre-invex F N V M  concide when Υ ~  is a lower UD pre-invex F N V M , see [84].
Assumption 1
([7]). Let us assume that K is an invex set with respect to the bi-function  φ : K × K R .  For any  ρ , μ K and  u 0 , 1 , so it yields
φ μ , ρ + u φ ( μ , ρ ) = 1 u φ μ , ρ ,
φ ρ , ρ + u φ ( μ , ρ ) = u φ μ , ρ .
For  u = 0, we have  φ μ , ρ = 0 if and only if  μ = ρ , and for all  ρ , μ K .
For more details on Assumption 1, please see references [83,84,85,86].

3. Main Results

We develop new results on the well-known fractional HH and HH–Fejér inequalities in the scope of UD pre-invex F N V M s. In the follow-up, we denote by L ρ , ρ + φ μ , ρ , 0 the family of Lebesgue measurable F N V M s.
Theorem 5.
Let Υ ~ : ρ , ρ + φ μ , ρ 0  be a UD pre-invex F N V M  on ρ , ρ + φ μ , ρ  whose b -levels define the family of I⋅V⋅Ms Υ b : ρ , ρ + φ μ , ρ R X o +  as Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b  for all  ϰ ρ , ρ + φ μ , ρ  and for all b 0 , 1 . If φ  satisfies Assumption 1 and Υ ~ L ρ , ρ + φ μ , ρ , 0 , then
Υ ~ 2 ρ + φ μ , ρ 2 F Γ β + 1 2 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ                         F Υ ~ ρ Υ ~ ρ + φ μ , ρ 2 F Υ ~ ρ Υ ~ μ 2 .
If  Υ ~ ( ϰ )  is UD pre-incave, then
Υ ~ 2 ρ + φ μ , ρ 2 F Γ β + 1 2 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ                         F Υ ~ ρ Υ ~ ρ + φ μ , ρ 2 F Υ ~ ρ Υ ~ μ 2 .
Proof. 
Let Υ ~ : ρ , ρ + φ μ , ρ 0 be a UD pre-invex F N V M . If Assumption 1 is verified, then we have
2 Υ ~ 2 ρ + φ μ , ρ 2 F Υ ~ ρ + 1 u φ μ , ρ Υ ~ ρ + u φ μ , ρ .
Therefore, for every b [ 0 , 1 ] , we have
2 Υ * 2 ρ + φ μ , ρ 2 , b Υ * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b , 2 Υ * 2 ρ + φ μ , ρ 2 , b Υ * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b .
By multiplying both sides by u β 1 and integrating the result with respect to u over ( 0,1 ) , it yields
2 0 1 u β 1 Υ * 2 ρ + φ μ , ρ 2 , b d u 0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b d u + 0 1 u β 1 Υ * ρ + u φ μ , ρ , b d u , 2 0 1 u β 1 Υ * 2 ρ + φ μ , ρ 2 , b d u 0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b d u + 0 1 u β 1 Υ * ρ + u φ μ , ρ , b d u .
Let ϰ = ρ + 1 u φ μ , ρ and y = ρ + u φ μ , ρ . Then, we get
2 β Υ * 2 ρ + φ μ , ρ 2 , b 1 φ μ , ρ β ρ ρ + φ μ , ρ ρ + φ μ , ρ y β 1 Υ * y , b d y + 1 φ μ , ρ β ρ ρ + φ μ , ρ ϰ ρ β 1 Υ * ϰ , b d ϰ Γ β φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b 2 β Υ * 2 ρ + φ μ , ρ 2 , b 1 φ μ , ρ β ρ ρ + φ μ , ρ ρ + φ μ , ρ y β 1 Υ * y , b d y + 1 φ μ , ρ β ρ ρ + φ μ , ρ ϰ ρ β 1 Υ * ϰ , b d ϰ Γ β φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b .
That is,
2 β Υ * 2 ρ + φ μ , ρ 2 , b , Υ * 2 ρ + φ μ , ρ 2 , b I Γ β φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b , I ρ + β Υ * ρ + φ μ , ρ , b + I μ β Υ * ρ + φ μ , ρ , b .
Thus,
2 β Υ ~ 2 ρ + φ μ , ρ 2 F Γ ( β ) φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ .
Adopting a similar procedure, we have
Γ ( β ) φ μ , ρ β [ I ρ + β Υ ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ ] F Υ ~ ρ Υ ~ ρ + φ μ , ρ 2 F Υ ~ ρ Υ ~ μ 2 .
Combining (37) and (38) yields
Υ ~ 2 ρ + φ μ , ρ 2 F Γ β + 1 2 φ μ , ρ β [ I ρ + β Υ ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ ] F Υ ~ ρ Υ ~ ρ + φ μ , ρ 2 F Υ ~ ρ Υ ~ μ 2 .
Thus, this concludes the proof. □
Remark 3.
If φ ϰ , y = ϰ y , then, from Theorem 5, we get [86]
Υ ~ ρ + μ 2 F Γ β + 1 2 μ ρ β I ρ + β Υ ~ μ I μ β Υ ~ ρ F Υ ~ ρ Υ ~ μ 2
Let us assume β = 1 . Then, Theorem 5 yields the result for a pre-invex F N V M [4]
Υ ~ 2 ρ + φ μ , ρ 2 F 1 φ μ , ρ ρ ρ + φ μ , ρ Υ ~ ϰ d ϰ F Υ ~ ρ Υ ~ μ 2
Let us assume β = 1 and φ ϰ , y = ϰ y . Then, Theorem 5 yields the result for convex F N V M [86]
Υ ~ ρ + μ 2 F 1 μ ρ ρ μ Υ ~ ϰ d ϰ F Υ ~ ρ Υ ~ μ 2
If Υ ~ is a lower UD pre-invex F N V M , then, from Theorem 5, we get [83]
Υ ~ 2 ρ + φ μ , ρ 2 F Γ β + 1 2 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ F Υ ~ ρ Υ ~ ρ + φ μ , ρ 2 F Υ ~ ρ Υ ~ μ 2 .
If Υ ~ is a lower UD pre-invex F N V M with φ ϰ , y = ϰ y , then, from Theorem 5, we get (see [83])
Υ ~ ρ + μ 2 F Γ β + 1 2 μ ρ β I ρ + β Υ ~ μ I μ β Υ ~ ρ F Υ ~ ρ Υ ~ μ 2
Let us consider β = 1 and Υ ~ a lower UD pre-invex F N V M . Then, Theorem 5 yields the result for pre-invex F N V M (see [84])
Υ ~ 2 ρ + φ μ , ρ 2 F 1 φ μ , ρ ρ ρ + φ μ , ρ Υ ~ ϰ d ϰ F Υ ~ ρ Υ ~ μ 2
Let us consider β = 1 and Υ ~ a lower UD pre-invex F⋅N⋅V⋅M  φ ϰ , y = ϰ y . Then, Theorem 5 yields the result for a convex F N V M (see [81])
Υ ~ ρ + μ 2 F 1 μ ρ ρ μ Υ ~ ϰ d ϰ F Υ ~ ρ Υ ~ μ 2
Let us assume that β = 1 = b and Υ * ( ϰ , b ) = Υ * ( ϰ , b ) with φ ϰ , y = ϰ y . Then, from Theorem 5, we obtain the classical HH–Fejér type inequality.
Example 1.
Let  β = 1 2 , ϰ 2,2 + φ 3,2  and the F N V M Υ ~ : ρ , ρ + φ μ , ρ = 2 , 2 + φ 3,2 0  be defined as
Υ ~ ϰ θ = θ 2 + ϰ 1 2 1 + ϰ 1 2       θ 2 ϰ 1 2 , 3 2 + ϰ 1 2 θ ϰ 1 2 + 1       θ ( 3 , 2 + ϰ 1 2 ] 0             o t h e r w i s e .
For each b 0 , 1 , we have Υ b ϰ = 1 b 2 ϰ 1 2 + 3 b , 1 b 2 + ϰ 1 2 + 3 b . Since the left and right end point functions Υ * ϰ , b = 1 b 2 ϰ 1 2 + 3 b and Υ * ϰ , b = 1 b 2 + ϰ 1 2 + 3 b are pre-invex functions with respect to φ μ , ρ = μ ρ , for each b [ 0 , 1 ] , Υ ~ ϰ is a UD pre-invex F N V M . We verify that Υ ~ L ρ , ρ + φ μ , ρ , 0 and
Υ * 2 ρ + φ μ , ρ 2 , b = Υ * 5 2 , b = 1 b 4 10 2 + 3 b , Υ * 2 ρ + φ μ , ρ 2 , b = Υ * 5 2 , b = 1 b 4 + 10 2 + 3 b , Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 = 1 b 4 2 3 2 + 3 b , Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 = 1 b 4 + 2 + 3 2 + 3 b .
Note that
Γ ( β + 1 ) 2 ( φ μ , ρ ) β I ρ + β Υ * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b = Γ ( 3 2 ) 2 1 π 2 2 + φ 3,2 3 ϰ 1 2 . 1 b 2 ϰ 1 2 + 3 b d ϰ + Γ ( 3 2 ) 2 1 π 2 2 + φ 3,2 ϰ 2 1 2 . 1 b 2 ϰ 1 2 + 3 b d ϰ         = 1 4 1 b 7393 10,000 + 9501 10,000 + 3 b    = 1 b 8447 20,000 + 3 b , Γ ( β + 1 ) 2 ( φ μ , ρ ) β I ρ + β Υ * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b      = Γ ( 3 2 ) 2 1 π 2 2 + φ 3,2 3 ϰ 1 2 . 1 b 2 + ϰ 1 2 + 3 b d ϰ      + Γ ( 3 2 ) 2 1 π 2 2 + φ 3,2 ϰ 2 1 2 . 1 b 2 + ϰ 1 2 + 3 b d ϰ      = 1 4 1 b 7260 1000 + 7049 1000 + 3 b = 1 b 14,309 4000 + 3 b .
Therefore,
1 b 4 10 2 + 3 b , 1 b 4 + 10 2 + 3 b I 1 b 8447 20,000 + 3 b , 1 b 14,309 4000 + 3 b I 1 b 4 2 3 2 + 3 b , 1 b 4 + 2 + 3 2 + 3 b ,
and Theorem 5 is verified.
We now introduce some new forms of fuzzy-interval fractional HH type inequalities for the product of UD pre-invex F N V M s, which are known as inequalities of Pachpatte type.
Theorem 6.
Let Υ ~ , J ~ : ρ , ρ + φ μ , ρ 0  be two UD pre-invex F N V M s on ρ , ρ + φ μ , ρ  whose b -levels Υ b , J b : ρ , ρ + φ μ , ρ R X o +  are defined by Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b  and J b ϰ = J * ϰ , b , J * ϰ , b  for all ϰ ρ , ρ + φ μ , ρ  and for all b 0 , 1 . If Υ ~ J ~ L ρ , ρ + φ μ , ρ , 0  and φ  satisfies Assumption 1, then
Γ β 2 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ J ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ J ~ ρ F 1 2 β ( β + 1 ) ( β + 2 ) B ~ ρ , ρ + φ μ , ρ β ( β + 1 ) ( β + 2 ) D ~ ρ , ρ + φ μ , ρ .
where  B ~ ρ , ρ + φ μ , ρ = Υ ~ ρ J ~ ρ Υ ~ ρ + φ μ , ρ J ~ ρ + φ μ , ρ ,   D ~ ρ , ρ + φ μ , ρ = Υ ~ ρ J ~ ρ + φ μ , ρ Υ ~ ρ + φ μ , ρ J ~ ρ ,   B b ρ , ρ + φ μ , ρ = B * ρ , ρ + φ μ , ρ , b , B * ρ , ρ + φ μ , ρ , b , and  D b ρ , μ = D * ρ , ρ + φ μ , ρ , b , D * ρ , ρ + φ μ , ρ , b .
Proof. 
Since Υ ~ , J ~ are UD pre-invex F N V M s and Assumption 1 holds f o r   φ , then for each b 0 , 1 we have
Υ * ρ + 1 u φ μ , ρ , b = Υ * ρ + φ μ , ρ + u φ ρ , ρ + φ μ , ρ , b u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b Υ * ρ + 1 u φ μ , ρ , b = Υ * ρ + φ μ , ρ + u φ ρ , ρ + φ μ , ρ , b u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b
and
J * ρ + 1 u φ μ , ρ , b = J * ρ + φ μ , ρ + u φ ρ , ρ + φ μ , ρ , b u J * ρ , b + 1 u J * ρ + φ μ , ρ , b J * ρ + 1 u φ μ , ρ , b = J * ρ + φ μ , ρ + u φ ρ , ρ + φ μ , ρ , b u J * ρ , b + 1 u J * ρ + φ μ , ρ , b .
From the UD pre-invex F N V M definition, we get 0 ~ F Υ ~ ϰ and 0 ~ F J ϰ . Therefore,
Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b u J * ρ , b + 1 u J * ρ + φ μ , ρ , b = u 2 Υ * ρ , b × J * ρ , b + 1 u 2 Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ + φ μ , ρ , b × J * ρ , b , Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b u J * ρ , b + 1 u J * ρ + φ μ , ρ , b = u 2 Υ * ρ , b × J * ρ , b + 1 u 2 Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ + φ μ , ρ , b × J * ρ , b .
Analogously, we have
Υ * ρ + u φ μ , ρ , b J * ρ + u φ μ , ρ , b 1 u 2 Υ * ρ , b × J * ρ , b + u 2 Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ + φ μ , ρ , b × J * ρ , b , Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b 1 u 2 Υ * ρ , b × J * ρ , b + u 2 Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ , b × J * ρ + φ μ , ρ , b + u 1 u Υ * ρ + φ μ , ρ , b × J * ρ , b .
Adding (48) and (49), we have
Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b u 2 + 1 u 2 Υ * ρ , b × J * ρ , b + Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + 2 u 1 u Υ * ρ + φ μ , ρ , b × J * ρ , b + Υ * ρ , b × J * ρ + φ μ , ρ , b , Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b u 2 + 1 u 2 Υ * ρ , b × J * ρ , b + Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + 2 u 1 u Υ * ρ + φ μ , ρ , b × J * ρ , b + Υ * ρ , b × J * ρ + φ μ , ρ , b .
Multiplying (50) by u β 1 and integrating the result with respect to u over (0,1), we obtain
0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + u β 1 Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b d u B * ρ , ρ + φ μ , ρ , b 0 1 u β 1 u 2 + 1 u 2 d u + 2 D * ρ , ρ + φ μ , ρ , b 0 1 u β 1 u 1 u d u , 0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + u β 1 Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b d u B * ρ , ρ + φ μ , ρ , b 0 1 u β 1 u 2 + 1 u 2 d u + 2 D * ρ , ρ + φ μ , ρ , b 0 1 u β 1 u 1 u d u .
It follows that
Γ β φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b × J * ρ , b 2 β 1 2 β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b + 2 β β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b , Γ β φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b × J * ρ , b 2 β 1 2 β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b + 2 β β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b ,
That is,
Γ β φ μ , ρ β [ I ρ + β Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b × J * ρ , b , I ρ + β Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b × J * ρ , b ] I 2 β 1 2 β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b , B * ρ , ρ + φ μ , ρ , b + 2 β β ( β + 1 ) ( β + 2 ) D * ρ , ρ + φ μ , ρ , b , D * ρ , ρ + φ μ , ρ , b .
Therefore,
Γ β 2 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ J ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ J ~ ρ F 1 2 β ( β + 1 ) ( β + 2 ) B ~ ρ , ρ + φ μ , ρ β β + 1 β + 2 D ~ ρ , ρ + φ μ , ρ ,
and, thus, the theorem is proven. □
Theorem 7.
Let Υ ~ , J ~ : ρ , ρ + φ μ , ρ 0  be two UD pre-invex F N V M s whose b -levels define the family of I V M s Υ b , J b : ρ , ρ + φ μ , ρ R X o +  given by Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b  and J b ϰ = J * ϰ , b , J * ϰ , b  for all ϰ ρ , ρ + φ μ , ρ  and for all b 0 , 1 . If Υ ~ J ~ L ρ , ρ + φ μ , ρ , 0  and φ  satisfies Assumption 1, then
1 β Υ ~ 2 ρ + φ μ , ρ 2 J ~ 2 ρ + φ μ , ρ 2 F Γ β + 1 4 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ J ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ J ~ ρ 1 2 β 1 2 β β + 1 β + 2 D ~ ρ , ρ + φ μ , ρ 1 2 β β β + 1 β + 2 B ~ ρ , ρ + φ μ , ρ ,
where  B ~ ρ , ρ + φ μ , ρ = Υ ~ ρ J ~ ρ Υ ~ ρ + φ μ , ρ J ~ ρ + φ μ , ρ ,   D ~ ρ , μ = Υ ~ ρ J ~ ρ + φ μ , ρ Υ ~ ρ + φ μ , ρ J ~ ρ ,   B b ρ , ρ + φ μ , ρ = B * ρ , ρ + φ ρ + φ μ , ρ , b , B * ρ , ρ + φ μ , ρ , b , and  D b ρ , ρ + φ μ , ρ = D * ρ , ρ + φ μ , ρ , b , D * ρ , ρ + φ μ , ρ , b .
Proof. 
Consider that Υ ~ , J ~ : ρ , ρ + φ μ , ρ 0 are UD pre-invex F N V M s . By hypothesization, for each b 0 , 1 , we have
Υ * 2 ρ + φ μ , ρ 2 , b × J * 2 ρ + φ μ , ρ 2 , b Υ * 2 ρ + φ μ , ρ 2 , b × J * 2 ρ + φ μ , ρ 2 , b 1 4 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + 1 u φ μ , ρ , b × J * ρ + u φ μ , ρ , b + 1 4 Υ * ρ + u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b 1 4 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + 1 u φ μ , ρ , b × J * ρ + u φ μ , ρ , b + 1 4 Υ * ρ + u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b , 1 4 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b + 1 4 u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b × 1 u J * ρ , b + u J * ρ + φ μ , ρ , b + 1 u Υ * ρ , b + u Υ * ρ + φ μ , ρ , b × u J * ρ , b + 1 u J * ρ + φ μ , ρ , b 1 4 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b + 1 4 u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b × 1 u J * ρ , b + u J * ρ + φ μ , ρ , b + 1 u Υ * ρ , b + u Υ * ρ + φ μ , ρ , b × u J * ρ , b + 1 u J * ρ + φ μ , ρ , b , = 1 4 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b + 1 4 u 2 + 1 u 2 D * ρ , ρ + φ μ , ρ , b + u 1 u + 1 u u B * ρ , ρ + φ μ , ρ , b , = 1 4 Υ * ρ + 1 u φ μ , ρ , b × J * ρ + 1 u φ μ , ρ , b + Υ * ρ + u φ μ , ρ , b × J * ρ + u φ μ , ρ , b + 1 4 u 2 + 1 u 2 D * ρ , ρ + φ μ , ρ , b + u 1 u + 1 u u B * ρ , ρ + φ μ , ρ , b .
Multiplying (52) with u β 1 and integrating over ( 0 , 1 ) , we get
1 β Υ * 2 ρ + φ μ , ρ 2 , b × J * 2 ρ + φ μ , ρ 2 , b 1 4 φ μ , ρ β ρ ρ + φ μ , ρ ρ + φ μ , ρ ϰ β 1 Υ * ϰ , b × J * ϰ , b d ϰ + ρ ρ + φ μ , ρ y ρ β 1 Υ * y , b × J * y , b d y + 1 2 β 1 2 β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b + 1 2 β β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b = Γ β + 1 4 φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ × J * ρ + φ μ , ρ + I ρ + φ μ , ρ β Υ * ρ × J * ρ + 1 2 β 1 2 β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b + 1 2 β β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b , 1 β Υ * 2 ρ + φ μ , ρ 2 , b × J * 2 ρ + φ μ , ρ 2 , b 1 4 φ μ , ρ β ρ ρ + φ μ , ρ ρ + φ μ , ρ ϰ β 1 Υ * ϰ , b × J * ϰ , b d ϰ + ρ ρ + φ μ , ρ y ρ β 1 Υ * y , b × J * y , b d y + 1 2 β 1 2 β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b + 1 2 β β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b = Γ β + 1 4 φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ × J * ρ + φ μ , ρ + I ρ + φ μ , ρ β Υ * ρ × J * ρ + 1 2 β 1 2 β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b + 1 2 β β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b ,
That is,
1 β Υ ~ 2 ρ + φ μ , ρ 2 × ~ J ~ 2 ρ + φ μ , ρ 2               F Γ β + 1 4 φ μ , ρ β I ρ + β Υ ~ ρ + φ μ , ρ J ~ ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ ρ J ~ ρ    1 2 β 1 2 β β + 1 β + 2 D ~ ρ , ρ + φ μ , ρ 1 2 β β β + 1 β + 2 B ~ ρ , ρ + φ μ , ρ .
Thus, this concludes the proof. □
Example 2.
Let  ρ , ρ + φ μ , ρ = [ 0 , φ 2,0 ] , β = 1 2 , Υ ~ ϰ = ϰ , 2 ϰ , and  J ~ ϰ = ϰ , 3 ϰ ,  as
Υ ~ ϰ θ = θ ϰ     θ 0 , ϰ 2 ϰ θ ϰ    θ ( ϰ , 2 ϰ ] 0     o t h e r w i s e ,
J ~ ϰ θ = θ 2 ϰ      θ ϰ , 2 4 ϰ θ 2 ϰ    θ ( 2 , ( 2 , 8 e ϰ ] ] 0      o t h e r w i s e .
Then, for each b 0 , 1 , we have Υ b ϰ = b ϰ , ( 2 b ) ϰ and J b ϰ = 1 b ϰ + 2 b , 1 b 8 e ϰ + 2 b . Since the left and right end point functions Υ * ϰ , b = b ϰ ,   Υ * ϰ , b = ( 2 b ) ϰ , J * ϰ , b = 1 b ϰ + 2 b , and J * ϰ , b = 1 b 8 e ϰ + 2 b are pre-invex functions with respect to φ μ , ρ = μ ρ and for each b [ 0 , 1 ] , then Υ ~ ϰ and J ~ ϰ are UD pre-invex F N V M s. We see that  Υ ~ ϰ J ~ ϰ L ρ , ρ + φ μ , ρ , 0 and
Γ 1 + β 2 φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b × J * ρ , b = Γ 3 2 2 2 1 π 0 φ 2,0 2 ϰ 1 2 b 1 b ϰ 2 + 2 b 2 ϰ d ϰ + Γ 3 2 2 2 1 π 0 φ 2,0 ϰ 1 2 b 1 b ϰ 2 + 2 b 2 ϰ d ϰ = 2 15 b 4 b + 11 , Γ 1 + β 2 φ μ , ρ β I ρ + β Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b + I ρ + φ μ , ρ β Υ * ρ , b × J * ρ , b = Γ 3 2 2 2 1 π 0 φ 2,0 2 ϰ 1 2 . 1 b 2 b ϰ 8 e ϰ + 2 b 2 b ϰ d ϰ + Γ 3 2 2 2 1 π 0 φ 2,0 ϰ 1 2 . 1 b 2 b ϰ 8 e ϰ + 2 b 2 b ϰ d ϰ = 2 5 2 b 8 3 b .
Note that
1 2 β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b = [ Υ * ρ , b × J * ρ , b + Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b ] = 22 15 b , 1 2 β β + 1 β + 2 B * ρ , ρ + φ μ , ρ , b = Υ * ρ , b × J * ρ , b + Υ * ρ + φ μ , ρ , b × J * ρ + φ μ , ρ , b = 11 15 . 2 b 1 b 8 e 2 + 2 b , β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b                                = Υ * ρ , b × J * ρ + φ μ , ρ , b + Υ * ρ + φ μ , ρ , b × J * ρ , b      = 8 15 b 2 , β β + 1 β + 2 D * ρ , ρ + φ μ , ρ , b                            = Υ * ρ , b × J * ρ + φ μ , ρ , b + Υ * ρ + φ μ , ρ × J * ρ , b         = 4 15 2 b 7 5 b .
Therefore, we have
1 2 β β + 1 β + 2 B b ρ , ρ + φ μ , ρ , b + β β + 1 β + 2 D b ρ , ρ + φ μ , ρ , b = 1 15 2 b 11 + 4 b , 2 b 11 1 b 8 e 2 + 2 b + 28 .
It follows that
1 5 [ 2 3 b 11 + 4 b , 2 2 b 8 3 b ] I 1 15 2 b 11 + 4 b , 2 b 11 1 b 8 e 2 + 2 b + 28
and Theorem 7 has been illustrated.
We now obtain second and first fuzzy fractional HH–Fejér type inequalities for UD pre-invex  F N V M .
Theorem 8.
Let Υ ~ : ρ , ρ + φ μ , ρ 0  be a UD pre-invex F N V M  with ρ < + φ μ , ρ  whose b -levels define the family of I V M Υ b : ρ , ρ + φ μ , ρ R X o +  given by Υ b ϰ = Υ * ϰ , b , Υ * ϰ , b  for all ϰ ρ , ρ + φ μ , ρ  and for all b 0 , 1 . Let Υ ~ L ρ , ρ + φ μ , ρ , 0  and C : ρ , ρ + φ μ , ρ R ,   C ( ϰ ) 0  be symmetric with respect to 2 ρ + φ μ , ρ 2 .  If φ  satisfies Assumption 1, then
I ρ + β Υ ~ C ρ + φ μ , ρ I ρ + φ μ , ρ β Υ ~ C ρ F Υ ~ ρ Υ ~ ρ + φ μ , ρ 2 I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ F Υ ~ ρ Υ ~ μ 2 I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ .
If  Υ ~ is UD pre-incave, then inequality (55) is reversed.
Proof. 
Let Υ ~ be a UD pre-invex F N V M and u β 1 C ρ + 1 u φ μ , ρ 0 . Then, for each b 0 , 1 , we have
u β 1 Υ * ρ + 1 u φ μ , ρ , b C ρ + 1 u φ μ , ρ u β 1 u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b C ρ + 1 u φ ρ + φ μ , ρ , ρ u β 1 Υ * ρ + 1 u φ μ , ρ , b C ρ + 1 u φ μ , ρ u β 1 u Υ * ρ , b + 1 u Υ * ρ + φ μ , ρ , b C ρ + 1 u φ ρ + φ μ , ρ , ρ ,
and
u β 1 Υ * ρ + u φ μ , ρ , b C ρ + u φ μ , ρ u β 1 1 u Υ * ρ , b + u Υ * ρ + φ μ , ρ , b C ρ + u φ μ , ρ u β 1 Υ * ρ + u φ μ , ρ , b C ρ + u φ μ , ρ u β 1 1 u Υ * ρ , b + u Υ * ρ + φ μ , ρ , b C ρ + u φ μ , ρ .
Adding (56) and (57), and integrating over 0 , 1 , we get
0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b C ρ + 1 u φ μ , ρ d u + 0 1 u β 1 Υ * ρ + u φ μ , ρ , b C ρ + u φ μ , ρ d u 0 1 u β 1 Υ * ρ , b u C ρ + 1 u φ μ , ρ + 1 u C ρ + u φ μ , ρ + u β 1 Υ * ρ + φ μ , ρ , b 1 u C ρ + 1 u φ μ , ρ + u C ρ + u φ μ , ρ d u , 0 1 u β 1 Υ * ρ + u φ μ , ρ , b C ρ + u φ μ , ρ d u + 0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b C ρ + 1 u φ μ , ρ d u 0 1 u β 1 Υ * ρ , b u C ρ + 1 u φ μ , ρ + 1 u C ρ + u φ μ , ρ + u β 1 Υ * ρ + φ μ , ρ , b 1 u C ρ + 1 u φ μ , ρ + u C ρ + u φ μ , ρ d u , = Υ * ρ , b 0 1 u β 1 C ρ + 1 u φ μ , ρ d u + Υ * ρ + φ μ , ρ , b 0 1 u β 1 C ρ + u φ μ , ρ d u , = Υ * ρ , b 0 1 u β 1 C ρ + 1 u φ μ , ρ d u + Υ * ρ + φ μ , ρ , b 0 1 u β 1 C ρ + u φ μ , ρ d u .
Since C is symmetric, then
= Υ * ρ , b + Υ * ρ + φ μ , ρ , b 0 1 u β 1 C ρ + u φ μ , ρ d u , = Υ * ρ , b + Υ * ρ + φ μ , ρ , b 0 1 u β 1 C ρ + u φ μ , ρ d u . = Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 Γ ( β ) ( φ μ , ρ ) β I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ , = Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 Γ ( β ) ( φ μ , ρ ) β I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ .
Since
0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b C ρ + u φ μ , ρ d u + 0 1 u β 1 Υ * ρ + u φ μ , ρ , b C ρ + u φ μ , ρ d u = 1 ( φ μ , ρ ) β ρ ρ + φ μ , ρ ϰ ρ β 1 Υ * 2 ρ + φ μ , ρ ϰ , b C ϰ d ϰ + 1 ( φ μ , ρ ) β ρ ρ + φ μ , ρ ϰ ρ β 1 Υ * ϰ , b C ϰ d ϰ = 1 ( φ μ , ρ ) β ρ ρ + φ μ , ρ ϰ ρ β 1 Υ * ϰ , b C 2 ρ + φ μ , ρ ϰ d ϰ + 1 ( φ μ , ρ ) β ρ ρ + φ μ , ρ ϰ ρ β 1 Υ * ϰ , b C ϰ d ϰ = Γ ( β ) ( φ μ , ρ ) β I ρ + β Υ * C μ + I μ β Υ * C ρ , 0 1 u β 1 Υ * ρ + 1 u φ μ , ρ , b C ρ + u φ μ , ρ d u + 0 1 u β 1 Υ * ρ + u φ μ , ρ , b C ρ + u φ μ , ρ d u = Γ ( β ) ( φ μ , ρ ) β I ρ + β Υ * C ρ + φ μ , ρ + I ρ + φ μ , ρ β Υ * C ρ .
Then, from (58), we have
Γ β φ μ , ρ β I ρ + β Υ * C ρ + φ μ , ρ + I ρ + φ μ , ρ β Υ * C ρ Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 Γ β φ μ , ρ β I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 Γ β φ μ , ρ β I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ , Γ β φ μ , ρ β I ρ + β Υ * C ρ + φ μ , ρ + I ρ + φ μ , ρ β Υ * C ρ Υ * ρ , b + Υ * ρ + φ μ , ρ , b 2 Γ β φ μ , ρ β I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ Υ * ρ , b + Υ * ρ + φ μ , b 2 Γ ( β ) φ μ , ρ β I ρ + β C ρ + φ μ , ρ + I ρ + φ μ , ρ β C ρ ,
That is,
Γ ( β ) φ μ , ρ β I ρ + β Υ * C ρ + φ μ , ρ + I ρ + φ μ , ρ β Υ * C ρ , I ρ + β Υ * C ρ + φ