Next Article in Journal
Creep Properties of a Viscoelastic 3D Printed Sierpinski Carpet-Based Fractal
Next Article in Special Issue
Lyapunov Functions and Stability Properties of Fractional Cohen–Grossberg Neural Networks Models with Delays
Previous Article in Journal
Mixed Fractional-Order and High-Order Adaptive Image Denoising Algorithm Based on Weight Selection Function
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels

by
Muhammad Bilal Khan
1,*,
Jorge E. Macías-Díaz
2,3,*,
Ali Althobaiti
4 and
Saad Althobaiti
5
1
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
2
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico
3
Department of Mathematics, School of Digital Technologies, Tallinn University, 3Narva Rd. 25, 10120 Tallinn, Estonia
4
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
5
Department of Sciences and Technology, Ranyah University Collage, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2023, 7(7), 567; https://doi.org/10.3390/fractalfract7070567
Submission received: 12 May 2023 / Revised: 9 July 2023 / Accepted: 17 July 2023 / Published: 24 July 2023
(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)

Abstract

:
The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ( U D E T - convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (HH-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for U D E T - convex   F N V M s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems.

1. Introduction

The convexity of functions is a strong technique used to solve a variety of pure and applied scientific problems. Many scientists have recently devoted themselves to investigating the properties and inequalities of convexity in various directions (see [1,2,3,4,5,6,7,8,9,10] and the references therein). H H inequality is one of the most important mathematical inequalities relevant to convex maps, and it is also frequently used in many other parts of practical mathematics, particularly in optimization and probability. Let us elicit it as follows.
The HH inequality for convex mapping Υ : K on an interval K = [ μ ,   b ] is
Υ ( μ + b 2 ) 1 b μ   μ b Υ ( ϰ ) d ϰ Υ ( μ ) + Υ ( b ) 2 ,
for all μ ,   b K ,   where K is a convex set.
This classical inequality gives error boundaries for the mean value of a continuous convex mapping Υ : K , which has piqued the interest of numerous scholars. Many investigations have been conducted on H H - type inequalities for additional forms of convex mappings. For example, Kórus [11] can be used to find s-convex mappings, Abramovich and Persson [12] for N-quasi-convex mappings, Delavar and De La Sen [13] for h-convex mappings, and so on. Ahmad et al. [14] created a fractional variation of the H H - type inequality by combining fractional integrals with exponential kernels. For more recent developments on this topic, see Khan et al. [15], Marinescu and Monea [16], şcan [17], Kadakal et al. [18], and Kadakal and Bekar [19], as well as [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] and the references therein.
As a very helpful tool, fractional calculus has proven to be a fundamental cornerstone in mathematics and applied sciences. Many researchers have been drawn to this field to investigate this issue. The H–H inequality for Reimann–Liouville fractional integrals [36], the H–H Fejér-type inequality for Katugampola fractional integrals [37], and extensions of trapezium inequalities for k-fractional integrals [38] are just a few examples of the prominent integral inequalities that have been studied by numerous authors through the successful interaction of different fractional calculus approaches. For further significant results relating to fractional integral operators, we direct interested readers to [39,40,41,42,43,44,45,46,47] and the references therein.
Set-valued analysis includes interval analysis. As shown by [48], interval analysis is unquestionably significant in both theoretical and applied research. Computing the error bounds of numerical finite-state machine solutions was one of the early uses of interval analysis. Over the past 50 years, however, interval analysis—one method for addressing interval uncertainty—has grown to be a crucial component of mathematical and computer models. Additionally, certain applications in computer graphics [49], automatic error analysis [50], and neural network output optimization [51] have been detailed here. Furthermore, [52,53,54,55,56,57,58,59,60,61,62,63] discussed several applications in optimization theory involving interval-valued functions. The interested reader is directed to Zhao et al. [64] and Román-Flores et al. [65] and the references therein for recent developments in association with interval-valued functions.
Khan and his colleagues recently extended the concept of convex interval-valued mappings (convex I∙V∙Ms) and fuzzy number-valued mappings by using fuzzy–order relations such as the convex F N V M s (apparently new) concept to include (h1, h2)-convex F N V M s (see [66]) and harmonic-convex F N V M s (see [67]). To demonstrate the inequalities of the H H - , H H - Fej'er, and Pachpatte types, his team used h-preinvex F N V M s [68], (h1, h2)-preinvex F N V M s [69], and higher-order preinvex F N V M s [70]. Khan et al. [71] recently developed novel H H - and H H Fej’er-type inequalities based on the recently introduced concept of fuzzy Reimann–Liouville fractional integrals via U D - F N V M s. We refer interested readers to investigate certain basic ideas connected to fuzzy calculus (see [72,73,74,75,76,77,78,79,80,81,82,83,84,85,86]) and the references therein for different recent breakthroughs linked to the notion of the fuzzy interval-valued analysis of several well-known integral inequalities.
In this study, we employed a novel fractional integral operator to provide more broadly applicable results. This is a result of the exponential kernel of this fuzzy fractional operator. Our findings differ from previous generalizations in that the aforementioned fractional analogous functional inequalities do not follow our conclusions. There is no exponential kernel in the results of experts’ expansions of the H H   inequality using different fuzzy fractional integral operators.
This work stimulated interest in the development of more generalized fuzzy fractional inequalities with an exponential kernel. A new direction in the study of inequalities is also introduced by the use of fuzzy number analysis in the key findings. To propose new inequalities, we have combined the ideas of a fuzzy analysis with a fuzzy U D   relation. There are still a lot of unanswered questions regarding fuzzy fractional integral inequalities involving different kinds of fuzzy convex number valued mappings, despite the fact that a lot of research has been done on the growth of fuzzy fractional order integral inequalities using convex fuzzy number valued functions. We develop new fuzzy H H - , Pachpatte, and Fej’er-type inequalities for U D E T - convex   F N V M s via fuzzy fractional integral operators having exponential kernels, which is this study’s main objective.

2. Preliminaries

Let X C be the space of all closed and bounded intervals of and Y X C be defined by
Y = [ Y * ,   Y * ] = { ϰ |   Y * ϰ Y * } ,   ( Y * ,   Y * ) .
If Y * = Y * , then Y is said to be degenerate. In this article, all intervals will be non-degenerate intervals. If Y * 0 , then [ Y * ,   Y * ] is called a positive interval. The set of all positive intervals is denoted by X C + and defined as
X C + = { [ Y * ,   Y * ] : [ Y * ,   Y * ] X C   and   Y * 0 } .
Let θ  and θ· Y be defined by
θ Y = { [ θ Y * ,   θ Y * ]   i f   θ > 0 , { 0 }             i f   θ = 0 , [ θ Y * , θ Y * ]     i f   θ < 0 .
Then, the Minkowski difference Z Y , addition Y + Z , and Y × Z for Y , Z X C are defined by
[ Z * ,   Z * ] + [ Y * ,   Y * ] = [ Z * + Y * ,     Z * + Y * ] ,
[ Z * ,   Z * ] × [ Y * ,   Y * ] = [ min { Z * Y * ,   Z * Y * ,   Z * Y * ,   Z * Y * } ,   max { Z * Y * ,   Z * Y * ,   Z * Y * ,   Z * Y * } ] ,
[ Z * ,   Z * ] [ Y * ,   Y * ] = [ Z * Y * ,     Z * Y * ] .
Remark 1.
 (i) For given   [ Z * ,   Z * ] ,   [ Y * ,   Y * ] X C ,  the relation   I  defined on  X C  by  [ Y * ,   Y * ] I [ Z * ,   Z * ]  if and only if  Y * Z * ,   Z * Y *  for all  [ Z * ,   Z * ] ,   [ Y * ,   Y * ] X C  is a partial interval inclusion relation. The relation  [ Y * ,   Y * ] I [ Z * ,   Z * ]  is coincident to  [ Y * ,   Y * ] [ Z * ,   Z * ]  on  X C .  It can be easily seen that “ I ” looks like “up and down” on the real line  ,  so we call  I  “up and down ” (or “ U D ” order, in short) [84](ii) For given  [ Z * ,   Z * ] ,   [ Y * ,   Y * ] X C ,  we say that  [ Z * ,   Z * ] I [ Y * ,   Y * ]  if and only if  Z * Y * ,   Z * Y *  or  Z * Y * ,   Z * < Y * , which is a partial interval order relation. The relation  [ Z * ,   Z * ] I [ Y * ,   Y * ]  is coincident to  [ Z * ,   Z * ] [ Y * ,   Y * ]  on   X C .  It can be easily seen that  I  looks likeleft and righton the real line  ,  so we call  I  “left and right(orLRorder, in short) [82,84]. For  [ Z * ,   Z * ] ,   [ Y * ,   Y * ] X C ,  the Hausdorff–Pompeiu distance between intervals  [ Z * ,   Z * ]  and  [ Y * ,   Y * ]  is defined by
d H ( [ Z * ,   Z * ] ,   [ Y * ,   Y * ] ) = m a x { | Z * Y * | ,   | Z * Y * | } .
It is a familiar fact that ( X C , d H ) is a complete metric space [37,40,41].
Note that, here, we are going to use the classical concepts of fuzzy set and fuzzy number so just we will recall some basic concepts related to fuzzy set and fuzzy numbers. Be aware that we designate the terms R and R C to signify, respectively, the set of all fuzzy subsets and fuzzy numbers of .
Definition 1 
 ([75,76]). Given  f ˜ R C  , the level sets or cut sets are given by  [ f ˜ ] θ = { ϰ |   f ˜ ( ϰ ) > θ }  for all  θ [ 0 ,   1 ]  and by  [ f ˜ ] 0 = { ϰ |   f ˜ ( ϰ ) > 0 } . These sets are known as  θ -level sets or   θ -cut sets of  f ˜ .
Proposition 1 
 ([82]). Let  f ˜ , g ˜ R C . Then, relation  F  is given on Rc by  f ˜ F g ˜  when and only when  [ f ˜ ] θ I [ g ˜ ] θ , for every  θ [ 0 ,   1 ] ,  which are left- and right-order relations.
Proposition 2 
([72]). Let  f ˜ , g ˜ R C . Then, relation  F  is given on  R C  by  f ˜ F g ˜  when and only when  [ f ˜ ] θ I [ g ˜ ] θ  for every  θ [ 0 ,   1 ] ,  which is  t h e   U D - order relation on  R C .
Remember the approaching notions, which are offered in the literature. If f ˜ , g ˜ R C and θ , then, for every θ [ 0 ,   1 ] , the arithmetic operations addition “ , multiplication “ , and scaler multiplication “ are defined by
[ f ˜ g ˜ ] θ = [ f ˜ ] θ + [ g ˜ ] θ ,
[ f ˜ g ˜ ] θ = [ f ˜ ] θ × [   g ˜ ] θ ,
[ 𝓉 f ˜ ] θ = 𝓉 ,
These operations follow directly from Equations (4)–(6), respectively.
Theorem 1 
 ([75]). The space  R C  dealing with a supremum metric, i.e., for  f ˜ ,   g ˜ R C
d ( f ˜ ,   g ˜ ) = sup 0 θ 1 d H ( [ f ˜ ] θ ,   [ g ˜ ] θ ) ,  
is a complete metric space, where  H   denotes the well-known Hausdorff metric on the space of intervals.

Fractional Integral Operators of Interval- and Fuzzy Number-Valued Mappings

Now, we define and discuss some properties of fractional integral operators for single-valued, interval-valued, and fuzzy number-valued mappings.
Theorem 2 
 ([75,79]). If  Υ : [ μ , b ] X C  is an interval-valued mapping (i-v-m) satisfying that  Υ ( ϰ ) = [ Υ * ( ϰ ) ,   Υ * ( ϰ ) ] , then  Υ  is an Aumann integrable (IA integrable) over  [ μ , b ]  when and only when  Υ * ( ϰ )  and  Υ * ( ϰ )  both are integrable over  [ μ , b ] ,  such that
( I A ) μ b Υ ( ϰ ) d ϰ = [ μ b Υ * ( ϰ ) d ϰ ,   μ b Υ * ( ϰ ) d ϰ ] .
Definition 2
 ([83]). Let  Υ ˜ : I R C be called F N V M . Then, for every  θ [ 0 ,   1 ] ,  as well as  θ  levels, define the family of i-v-ms  Υ θ : I X C ,  satisfying that  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ]  for every  ϰ I .  Here, for every  θ [ 0 ,   1 ] ,  the end point real valued mappings  Υ * ( · , θ ) ,   Υ * ( · , θ ) : I  are called lower and upper mappings of  Υ .
Definition 3
 ([83]). Let  Υ ˜ : I R C  be a  F N V M . Then,  Υ ˜ ( ϰ )  is said to be continuous at  ϰ I  if, for every  θ [ 0 ,   1 ] ,   Υ θ ( ϰ ) ,   i t  is continuous when and only when both end point mappings  Υ * ( ϰ , θ ) and Υ * ( ϰ , θ )  are continuous at  ϰ I .
Definition 4
 ([79]). Let  Υ ˜ : [ μ ,   b ] R C  be  F N V M . The fuzzy Aumann integral ( ( F A )  integral) of  Υ ˜  over  [ μ ,   b ] ,  denoted by  ( F A ) μ b Υ ˜ ( ϰ ) d ϰ , is defined level-wise by
F A μ b Υ ~ ϰ d ϰ   θ = I A μ b Υ θ ϰ d ϰ = μ b Υ ϰ , θ d ϰ : Υ ϰ , θ S Υ θ ,
where  S ( Υ θ ) = { Υ ( . , θ ) : Υ ( . , θ )   i s   i n t e g r a b l e   a n d   Υ ( ϰ , θ ) Υ θ ( ϰ ) }   for every  θ [ 0 ,   1 ] . Υ ˜  is the  ( F A )  integrable over  [ μ ,   b ]  if  ( F A ) μ b Υ ˜ ( ϰ ) d ϰ R C .
Theorem 3
 ([80]). Let  Υ : [ μ ,   b ] R C  be a  F N V M ,  as well as  θ  levels, define the family of i-v-ms  Υ θ : [ μ ,   b ] X C , satisfying that  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ] for every ϰ [ μ ,   b ]  and for every  θ [ 0 ,   1 ] .  Then, Υ is  ( F A )  integrable over  [ μ ,   b ]  when and only when,  Υ * ( ϰ , θ ) and Υ * ( ϰ , θ )  both are integrable over  [ μ ,   b ] . Moreover, if  Υ  is  ( F A )  integrable over  [ μ ,   b ] ,  then
[ ( F A ) μ b Υ ( ϰ ) d ϰ ]   θ = [ μ b Υ * ( ϰ , θ ) d ϰ ,   μ b Υ * ( ϰ , θ ) d ϰ ] = ( I A ) μ b Υ θ ( ϰ ) d ϰ ,
for every θ∈[0,1].
Allahviranloo et al. [74] introduced the following fuzzy Reimann–Liouville fractional integral operators:
Definition 5.
Let  α > 0 and L ( [ μ ,   b ] , R C )  be the collection of all Lebesgue measurable  F N V M s  on  [ μ , b ] . Then, the fuzzy left and right Reimann–Liouville fractional integrals of  Υ ˜   L ( [ μ ,   b ] , R C )  with order  α > 0  are defined by
μ + α Υ ˜ ( ϰ ) = 1 Γ ( α ) μ ϰ ( ϰ 𝓉 ) α 1 Υ ˜ ( 𝓉 ) d 𝓉 ,   ( ϰ > μ ) ,  
and
b α Υ ˜ ( ϰ ) = 1 Γ ( α ) ϰ b ( 𝓉 ϰ ) α 1 Υ ˜ ( 𝓉 ) d 𝓉 ,   ( ϰ < b ) ,
respectively, where  Γ ( ϰ ) = 0 𝓉 ϰ 1 e 𝓉 d 𝓉   is the Euler gamma mapping. The fuzzy left and right Reimann–Liouville fractional integral  ϰ  based on left and right end point mappings can be defined as
[ μ + α   Υ ˜ ( ϰ ) ] θ = 1 Γ ( α ) μ ϰ ( ϰ 𝓉 ) α 1 Υ θ ( 𝓉 ) d 𝓉 = 1 Γ ( α ) μ ϰ ( ϰ 𝓉 ) α 1 [ Υ * ( 𝓉 ,   θ ) , Υ * ( 𝓉 ,   θ ) ] d 𝓉 ,   ( ϰ > μ ) ,  
where
μ + α   Υ * ( ϰ ,   θ ) = 1 Γ ( α ) μ ϰ ( ϰ 𝓉 ) α 1 Υ * ( 𝓉 ,   θ ) d 𝓉 ,   ( ϰ > μ ) ,
and
μ + α   Υ * ( ϰ ,   θ ) = 1 Γ ( α ) μ ϰ ( ϰ 𝓉 ) α 1 Υ * ( 𝓉 ,   θ ) d 𝓉 ,   ( ϰ > μ ) ,
Similarly, we can define the right Reimann–Liouville fractional integral Υ of ϰ based on the left and right end point mappings.
Definition 6
 ([73]). The mapping of  Υ : [ μ ,   b ]  is called exponential trigonometric convex mapping on    [ μ ,   b ]  if
Υ ( 𝓉 ϰ + ( 1 𝓉 ) s ) s i n π 𝓉 2 e 1 𝓉 Υ ( ϰ ) c o s π 𝓉 2 e 𝓉 Υ ( s ) ,  
for all   ϰ ,   s [ μ ,   b ] ,   𝓉 [ 0 ,   1 ] ,  and  ϰ [ μ ,   b ] .  If (21) is reversed, then  Υ  is called exponential trigonometric concave mapping on [μ,b].
Definition 7
 ([85]).  F N V M   Υ ˜ : [ μ ,   b ] R C  is called UD–convex FNVM on   [ μ ,   b ]  if
Υ ˜ ( 𝓉 ϰ + ( 1 𝓉 ) s   ) F 𝓉 Υ ˜ ( ϰ ) ( 1 𝓉 ) Υ ˜ ( s ) ,
for all   ϰ ,   s [ μ ,   b ] ,   𝓉 [ 0 ,   1 ] ,  where  Υ ˜ ( ϰ ) F 0 ˜  for all  ϰ [ μ ,   b ] .  If (22) is reversed, then  Υ ˜  is called concave  F N V M  on  [ μ ,   b ] . Υ ˜  is affinal if and only if it is both convex and concave  F N V M .
Definition 8
 ([86]). The  F N V M Υ ˜ : [ μ ,   b ] R C  is called convex  F N V M on   [ μ ,   b ]  if
Υ ˜ ( 𝓉 ϰ + ( 1 𝓉 ) s ) F 𝓉 Υ ˜ ( ϰ ) ( 1 𝓉 ) Υ ˜ ( s ) ,  
for all   ϰ ,   s [ μ ,   b ] ,   𝓉 [ 0 ,   1 ] ,  where  Υ ˜ ( ϰ ) F 0 ˜  for all  ϰ [ μ ,   b ] .  If (23) is reversed, then  Υ ˜  is called  U D concave  F N V M  on  [ μ ,   b ] . Υ ˜  is  U D - affine   F N V M  if and only if it is both  U D - convex and UD-concave FNVM.
Theorem 4
 ([86]). Let  Υ ˜ : [ μ , b ] R C  be a   F N V M , with  θ  levels defined by the family of interval-valued mappings (i-v-ms)  Υ θ : [ μ , b ] X C + X C  given by
Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ] ,    
for all  ϰ [ μ , b ]  and for all  θ [ 0 ,   1 ] . Then,  Υ ˜  is  U D - c o n v e x   F N V M  on  [ μ , b ]  if and only if, for all  θ [ 0 ,   1 ] ,   Υ * ( ϰ ,   θ )  is a convex mapping and  Υ * ( ϰ ,   θ )  is a concave mapping.
Definition 9
 ([74]). Let  α > 0  and  L ( [ μ ,   b ] ,   )  be the collection of all Lebesgue measurable mappings on  [ μ , b ] . Then, the left and right Reimann–Liouville fractional integrals with exponential kernels in connection of ΥL ([μ,b], ℝ) with order  α > 0  are defined by
μ + α   Υ ( ϰ ) = 1 α μ ϰ e ( 1 α α ( ϰ 𝓉 ) ) Υ ( 𝓉 ) d 𝓉 ,         ( ϰ > μ ) ,  
and
b α   Υ ( ϰ ) = 1 α ϰ b e ( 1 α α ( 𝓉 ϰ ) ) Υ ( 𝓉 ) d 𝓉 ,   ( ϰ < b ) ,  
respectively.
Now, we present newly defined the fuzzy left and right Reimann–Liouville fractional integral with exponential kernels with respect to F N V M   Υ ˜   L ( [ μ ,   b ] , R C ) .
Definition 10
 ([78]). Let  Υ ˜ : [ μ ,   b ] R C  be  F N V M , and Υ ˜  is the fuzzy Aumann integrable over  [ μ ,   b ] .  Then, the fuzzy left and right Reimann–Liouville fractional integrals with exponential kernels in connection of  Υ ˜   L ( [ μ ,   b ] , R C )  with order  α > 0  are defined by
[ μ + α   Υ ˜ ( ϰ ) ]   θ = μ + α Υ θ ( ϰ ) = { μ + α Υ ( ϰ , θ ) : Υ ( ϰ , θ ) S ( Υ θ ) } ,  
and
[ b α   Υ ˜ ( ϰ ) ]   θ = b α Υ θ ( ϰ ) = { b α Υ ( ϰ , θ ) : Υ ( ϰ , θ ) S ( Υ θ ) } ,
where  S ( Υ θ ) =  { Υ ( . , θ ) : Υ ( . , θ )  is integrable and Υ(ϰ,θ)∈Υθ(ϰ)} for every θ∈ [0,1].  S ( Υ θ ) =  { Υ ( . , θ ) : Υ ( . , θ )  are the left and right Reimann–Liouville fractional integrable with exponential kernels and order  α > 0  for every  θ [ 0 ,   1 ] }.  Υ ˜  is fuzzy left and right Reimann–Liouville fractional integrals with exponential Kernels and order  α > 0  over  [ μ ,   b ]  if  μ + α   Υ ˜ ( ϰ ) ,   b α Υ ˜ ( ϰ ) R C .
Definition 11
 ([78]). Let  α > 0 and L ( [ μ ,   b ] , R C )  be the collection of all Lebesgue measurable  F N V M on [ μ , b ] . Then, the fuzzy left and right Reimann–Liouville fractional integrals with exponential kernels in connection with  Υ ˜ L ( [ μ , b ] , R C ) with order  α > 0  are defined by
μ + α Υ ˜ ( ϰ ) = 1 α μ ϰ e ( 1 α α ( ϰ 𝓉 ) ) Υ ˜ ( 𝓉 ) d 𝓉 ,   ( ϰ > μ ) ,  
and
b α Υ ˜ ( ϰ ) = 1 α ϰ b e ( 1 α α ( 𝓉 ϰ ) ) Υ ˜ ( 𝓉 ) d 𝓉 ,   ( ϰ < b ) ,
respectively. The fuzzy left Reimann–Liouville fractional integrals with exponential kernels in connection with left and right end point mappings can be defined as
[ μ + α   Υ ˜ ( ϰ ) ] θ = 1 α μ ϰ e ( 1 α α ( ϰ 𝓉 ) ) Υ θ ( 𝓉 ) d 𝓉   = 1 α μ ϰ e ( 1 α α ( ϰ 𝓉 ) ) [ Υ ( 𝓉 , θ ) , Υ ( 𝓉 , θ ) ] d 𝓉 , ( ϰ > μ ) ,
where
μ + α   Υ * ( ϰ ,   θ ) = 1 α μ ϰ e ( 1 α α ( ϰ 𝓉 ) ) Υ * ( 𝓉 ,   θ ) d 𝓉 ,   ( ϰ > μ ) ,
and
μ + α   Υ * ( ϰ ,   θ ) = 1 α μ ϰ e ( 1 α α ( ϰ 𝓉 ) ) Υ * ( 𝓉 ,   θ ) d 𝓉 ,   ( ϰ > μ ) .
Similarly, the fuzzy right Reimann–Liouville fractional integral with exponential kernels in connection of left and right end point mappings can be defined as
[ b α   Υ ˜ ( ϰ ) ] θ = 1 α ϰ b e ( 1 α α ( 𝓉 ϰ ) ) Υ θ ( 𝓉 ) d 𝓉 ,     = 1 α ϰ b e ( 1 α α ( 𝓉 ϰ ) ) [ Υ * ( 𝓉 ,   θ ) , Υ * ( 𝓉 ,   θ ) ] d 𝓉 ,         ( ϰ < b )
where
b α   Υ * ( ϰ ,   θ ) = 1 α ϰ b e ( 1 α α ( 𝓉 ϰ ) ) Υ * ( 𝓉 ,   θ ) d 𝓉 ,         ( ϰ < b )
and
b α   Υ * ( ϰ ,   θ ) = 1 α ϰ b e ( 1 α α ( 𝓉 ϰ ) ) Υ * ( 𝓉 ,   θ ) d 𝓉 ,         ( ϰ < b ) .
Now, we introduce the new class of convex F N V M , which is known as U D E T - convex   F N V M .
Definition 12.
The  F N V M   Υ ˜ : [ μ ,   b ] R C is called U D E T - convex F N V M  on   [ μ ,   b ]  if
Υ ˜ ( 𝓉 ϰ + ( 1 𝓉 ) s ) F s i n π 𝓉 2 e 1 𝓉 Υ ˜ ( ϰ ) c o s π 𝓉 2 e 𝓉 Υ ˜ ( s ) ,  
for all   ϰ ,   s [ μ ,   b ] ,   𝓉 [ 0 ,   1 ] ,  where  Υ ˜ ( ϰ ) F 0 ˜  for all  ϰ [ μ ,   b ] .  If (35) is reversed, then  Υ ˜  is called  U D E T - c o n c a v e   F N V M  on  [ μ ,   b ] .
Theorem 5.
Let  K  be an invex set and  Υ ˜ : K R C  be a  F N V M , with  θ  levels defined by the family of IVFs  Υ θ : K K C + K C  given by
Υ θ ( x ) = [ Υ * ( x , θ ) ,   Υ * ( x , θ ) ] ,     x K .
for all  x K  and for all  θ [ 0 ,   1 ] . Then,  Υ ˜  is the  U D E T - convex  F N V M  on  K  if and only if, for all  θ [ 0 ,   1 ] ,   Υ * ( x ,   θ )  and  Υ * ( x ,   θ )  are exponential trigonometric convex and concave functions, respectively.
Proof. 
Assume that, for each θ [ 0 ,   1 ] ,   Υ * ( x ,   θ ) and Υ * ( x ,   θ ) are exponential trigonometric convex and concave functions on K , respectively. Then, from (21), we have
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) sin π 𝓉 2 e 1 𝓉 Υ * ( x ,   θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s ,   θ ) ,   x , s K ,   𝓉 [ 0 ,   1 ] ,
and
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) sin π 𝓉 2 e 1 𝓉 Υ * ( x ,   θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s ,   θ ) ,   x , s K ,   𝓉 [ 0 ,   1 ] .
Then, using (36), (5), and (7), we obtain
Υ θ ( 𝓉 ϰ + ( 1 𝓉 ) s )                       = [ Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) ,   Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) ] ,               I sin π 𝓉 2 e 1 𝓉 [ Υ * ( x ,   θ ) ,   Υ * ( x ,   θ ) ] + cos π 𝓉 2 e 𝓉 [ Υ * ( s ,   θ ) ,   Υ * ( s ,   θ ) ] ,            
which is
Υ ˜ ( 𝓉 ϰ + ( 1 𝓉 ) s ) F sin π 𝓉 2 e 1 𝓉 Υ ˜ ( x ) cos π 𝓉 2 e 𝓉 Υ ˜ ( s ) ,   x , s K ,   𝓉 [ 0 ,   1 ] .
Hence, Υ ˜ is the U D E T - convex F N V M on K .
Conversely, let Υ ˜ be a U D E T convex F N V M on K . Then, for all x , s K and 𝓉 [ 0 ,   1 ] , we have
Υ ˜ ( 𝓉 ϰ + ( 1 𝓉 ) s ) F sin π 𝓉 2 e 1 𝓉 Υ ˜ ( x ) cos π 𝓉 2 e 𝓉 Υ ˜ ( s ) .
Therefore, from (36), we have
Υ θ ( 𝓉 ϰ + ( 1 𝓉 ) s ) = [ Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) ,   Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) ] .
Again, from (36), (5), and (7), we obtain
sin π 𝓉 2 e 1 𝓉 Υ θ ( x ) + cos π 𝓉 2 e 𝓉 Υ θ ( x ) = [ sin π 𝓉 2 e 1 𝓉 Υ * ( x ,   θ ) ,   sin π 𝓉 2 e 1 𝓉 Υ * ( x ,   θ ) ] + [ cos π 𝓉 2 e 𝓉 Υ * ( s ,   θ ) ,   cos π 𝓉 2 e 𝓉 Υ * ( s ,   θ ) ] ,
for all x , s K and 𝓉 [ 0 ,   1 ] . Then, by U D E T - convexity of Υ ˜ , we have, for all, x,sK and 𝓉 [ 0 ,   1 ] ,   such that
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) sin π 𝓉 2 e 1 𝓉 Υ * ( x ,   θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s ,   θ ) ,
and
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) sin π 𝓉 2 e 1 𝓉 Υ * ( x ,   θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s ,   θ ) ,
for each θ [ 0 ,   1 ] . Hence, the results follow. □
Remark 2.
If  Υ * ( ϰ , θ ) Υ * ( ϰ , θ )  and  θ = 1 , then we obtain the definition of an exponential trigonometric convex interval-valued function, which is also a new one:
Υ ( 𝓉 ϰ + ( 1 𝓉 ) s ) s i n π 𝓉 2 e 1 𝓉 Υ ( ϰ ) + c o s π 𝓉 2 e 𝓉 Υ ( s ) .  
If  Υ * ( ϰ , θ ) = Υ * ( ϰ , θ )  and  θ = 1 , then we obtain the classical definition of exponential trigonometric convex functions.
Here are some new definitions that will support us in acquiring some classical results.
Definition 13.
Let  Υ ˜ : [ μ , b ] R C  be a  F N V M , with  θ  levels defined by the family of I-V∙Ms  Υ θ : [ μ , b ] X C + X C  given by
Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ] ,    
for all  ϰ [ μ , b ]  and for all  θ [ 0 ,   1 ] . Then,  Υ ˜  is a lower  U D    exponential trigonometric convex (concave)   F N V M  on  [ μ , b ]  if and only if, for all,   θ [ 0 ,   1 ] ,
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) ( ) sin π 𝓉 2 e 1 𝓉 Υ * ( ϰ , θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s , θ ) ,
and
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) = sin π 𝓉 2 e 1 𝓉 Υ * ( ϰ , θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s , θ ) .
Definition 14.
Let  Υ ˜ : [ μ , b ] R C  be a  F N V M , with   θ  levels defined by the family of I-V∙Ms  Υ θ : [ μ , b ] X C + X C  given by
Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ] ,  
for all  ϰ [ μ , b ]  and for all  θ [ 0 ,   1 ] . Then,   Υ ˜  is the  U D    exponential trigonometric convex (concave)  F N V M  on if and only if, for all,  θ [ 0 ,   1 ] ,
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) = sin π 𝓉 2 e 1 𝓉 Υ * ( ϰ , θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s , θ ) ,
and
Υ * ( 𝓉 ϰ + ( 1 𝓉 ) s ,   θ ) ( ) sin π 𝓉 2 e 1 𝓉 Υ * ( ϰ , θ ) + cos π 𝓉 2 e 𝓉 Υ * ( s , θ ) .
Remark 3.
Both concepts “ U D E T - convex  F N V M ” and “exponential trigonometric convex  F N V M ” behave alike when  Υ ˜  is  U D E T - convex  F N V M .

3. Main Results

In this part, we will suggest a novel version of fuzzy H–H inequalities, and we will use nontrivial examples to demonstrate how these fuzzy up and down relations work.
Theorem 6.
Let  Υ ˜ : [ μ ,   b ] R C  be an  U D E T - convex  F N V M  on  [ μ ,   b ] ,  with  θ  levels defined by the family of i-v-ms  Υ θ : [ μ ,   b ] X C +  given by  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ]  for all  ϰ [ μ ,   b ]  and for all  θ [ 0 ,   1 ] . If  Υ ˜ L ( [ μ ,   b ] , R C ) , then
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν ) [ μ + α   Υ ˜ ( b ) b α   Υ ˜ ( μ ) ] F ν 1 e ν S ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .  
If Υ ˜ ( ϰ )  is  U D E T concave  F N V M , then
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν ) [ μ + α Υ ˜ ( b ) b α Υ ˜ ( μ ) ] F ν 1 e ν S ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
where
S ( ν ) = 4 ν + 2 π e ν 1 + 4 4 ν 2 + 8 ν + π 2 + 4 + 2 π e 1 + 4 e ν 1 ( e + ν e ) 4 ν 2 8 ν + π 2 + 4 , ν = 1 α α ( b μ )   and   1 > α > 0
Proof. 
Let Υ ˜ : [ μ ,   b ] R C be an U D E T - convex F N V M . Then, by hypothesis, we have
Υ ˜ ( μ + b 2 ) F sin π 4 e Υ ˜ ( 𝓉 μ + ( 1 𝓉 ) b ) cos π 4 e Υ ˜ ( ( 1 𝓉 ) μ + 𝓉 b ) .
After simplification, we get that
2 Υ ˜ ( μ + b 2 ) F 2 e [ Υ ˜ ( 𝓉 μ + ( 1 𝓉 ) b ) Υ ˜ ( ( 1 𝓉 ) μ + 𝓉 b ) ] .
Therefore, for every θ [ 0 ,   1 ] , we have
2 Υ * ( μ + b 2 ,   θ ) 2 e [ Υ * ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) + Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) ] ,
2 Υ * ( μ + b 2 ,   θ ) 2 e [ Υ * ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) + Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) ] .
Taking Υ * ( . ,   θ ) and multiplying both sides by e ν 𝓉 and integrating the obtained results with respect to 𝓉 from 0 to 1 , we have
2 0 1 e ν 𝓉 Υ * ( μ + b 2 ,   θ ) d 𝓉 2 e [ 0 1 e ν 𝓉 Υ * ( 𝓉 μ + ( 1 𝓉 ) b , θ ) d 𝓉 + 0 1 e ν 𝓉 Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) d 𝓉 ] .    
Let u = 𝓉μ+ (1−𝓉)b and x = ( 1 𝓉 ) μ + 𝓉 b . Then, we have
2 0 1 e ν 𝓉 Υ * ( μ + b 2 ,   θ ) d 𝓉 2 e 1 b μ   μ b e ( 1 α α ( b u ) ) Υ * ( u , θ ) d u + 1 b μ μ b e ( 1 α α ( x μ ) ) Υ * ( ϰ , θ ) d ϰ   = 2 e α b μ [ μ + α   Υ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) ] .
Now, taking the right side of Equation (42), we have
0 1 e ν 𝓉 Υ * ( μ + b 2 ,   θ ) d 𝓉 = 1 e ν ν Υ * ( μ + b 2 ,   θ )
From (42) and (43), we have
2 α · 1 e ν ν Υ * ( μ + b 2 ,   θ ) 2 e · 1 b μ [ μ + α   Υ * ( b ,   θ ) + b α Υ * ( μ ,   θ ) ] .
Similarly, for Υ * ( ϰ , θ ) , we have
2 α · 1 e ν ν Υ * ( μ + b 2 ,   θ ) 2 e · 1 b μ [ μ + α   Υ * ( b ,   θ ) + b α Υ * ( μ ,   θ ) ] .
From (44) and (45), we have
2 α · 1 e ν ν [ Υ * ( μ + b 2 ,   θ ) , Υ * ( μ + b 2 ,   θ ) ] I 2 e · 1 b μ [ [ μ + α   Υ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) ] ,   [ μ + α   Υ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) ] ] .
That is
2 α · 1 e ν ν   Υ θ ( μ + b 2 ) I 2 e · 1 b μ [ μ + α   Υ θ ( b ) + b α   Υ θ ( μ ) ] .
For the right side of Equation (40), since Υ ˜ is an U D E T - convex F N V M , then we deduce that
Υ ˜ ( 𝓉 μ + ( 1 𝓉 ) b ) F sin π 𝓉 2 e 1 𝓉 Υ ˜ ( μ ) cos π 𝓉 2 e 𝓉 Υ ˜ ( b ) ,
and
Υ ˜ ( ( 1 𝓉 ) μ + 𝓉 b   ) F cos π 𝓉 2 e 𝓉 Υ ˜ ( μ ) sin π 𝓉 2 e 1 𝓉 Υ ˜ ( b ) .
Adding (47) and (48), we have
Υ ˜ ( 𝓉 μ + ( 1 𝓉 ) b   ) Υ ˜ ( ( 1 𝓉 ) μ + 𝓉 b   ) F [ Υ ˜ ( μ ) Υ ˜ ( b ) ] [ sin π 𝓉 2 e 1 𝓉 + cos π 𝓉 2 e 𝓉 ] .
Since Υ ~ is F N V M , then, for each   θ [ 0 ,   1 ] , we have
Υ * ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) + Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] [ sin π 𝓉 2 e 1 𝓉 + cos π 𝓉 2 e 𝓉 ] ,   Υ * ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) + Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] [ sin π 𝓉 2 e 1 𝓉 + cos π 𝓉 2 e 𝓉 ] .  
Taking Υ * ( . ,   θ ) from (50) and multiplying the inequality with e ν 𝓉 and integrating the results with 𝓉 from 0 to 1 , we have
0 1 e ν 𝓉 Υ * ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) d 𝓉 + 0 1 e ν 𝓉 Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) d 𝓉 [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] 0 1 e ν 𝓉 [ sin π 𝓉 2 e 1 𝓉 + cos π 𝓉 2 e 𝓉 ] d 𝓉 , = 4 ν + 2 π e ν 1 + 4 4 ν 2 + 8 ν + π 2 + 4 + 2 π e 1 + 4 e ν 1 ( e + ν e ) 4 ν 2 8 ν + π 2 + 4 [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] .
In a manner identical to that described previously, we have Υ * ( . ,   θ )
0 1 e ν 𝓉 Υ * ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) d 𝓉 + 0 1 e ν 𝓉 Υ * ( ( 1 𝓉 ) μ + 𝓉 b ,   θ ) d 𝓉 [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] 0 1 e ν 𝓉 [ sin π 𝓉 2 e 1 𝓉 + cos π 𝓉 2 e 𝓉 ] d 𝓉 , = 4 ν + 2 π e ν 1 + 4 4 ν 2 + 8 ν + π 2 + 4 + 2 π e 1 + 4 e ν 1 ( e + ν e ) 4 ν 2 8 ν + π 2 + 4 [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] .
From (51) and (52), we have
0 1 e ν 𝓉 Υ θ ( 𝓉 μ + ( 1 𝓉 ) b   ) d 𝓉 + 0 1 e ν 𝓉 Υ θ ( ( 1 𝓉 ) μ + 𝓉 b   ) d 𝓉 I 4 ν + 2 π e ν 1 + 4 4 ν 2 + 8 ν + π 2 + 4 + 2 π e 1 + 4 e ν 1 ( e + ν e ) 4 ν 2 8 ν + π 2 + 4 [ Υ θ ( μ ) + Υ θ ( b ) ] .
From (46) and (53), we have
e 2 Υ θ ( μ + b 2 ) I 1 α 2 ( 1 e ν ) [ μ + α   Υ θ ( b ) + b α   Υ θ ( μ ) ] I ν 1 e ν S ( ν ) Υ θ ( μ ) + Υ θ ( b ) 2 ,
that is
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν ) [ μ + α   Υ ˜ ( b ) b α   Υ ˜ ( μ ) ] F ν 1 e ν S ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
Hence, the required results. □
If we use a few minor limits on Theorem 6, then the novel and conventional results can be as follows:
Remark 4.
We readily see, from Theorem 6, that
If Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then we have (see [78])
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν ) [ μ + α Υ ˜ ( b ) b α Υ ˜ ( μ ) ] F ν 1 e ν S ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .  
Let Υ * ( ϰ , θ ) Υ * ( ϰ , θ ) and θ = 1 , . Then, from Theorem 6, we have (see [78])
e 2 Υ ( μ + b 2 ) 1 α 2 ( 1 e ν ) [ μ + α   Υ ( b ) + b α   Υ ( μ ) ] ν 1 e ν S ( ν ) Υ ( μ ) + Υ ( b ) 2 .
If α 1 , then
lim α 1 ν = lim α 1 1 α α ( b μ ) = 0 ,   then
lim α 1 4 ν + 2 π e ν 1 + 4 4 ν 2 + 8 ν + π 2 + 4 + 2 π e 1 + 4 e ν 1 e + ν e 4 ν 2 8 ν + π 2 + 4 = 2 π e 1 + 4 π 2 + 4 ,   lim α 1 1 α 2 1 e ν = 1 2 b μ
Now, from Theorem 6, we acquire the following result, which is also new one:
e 2 Υ ˜ ( μ + b 2 ) F 1 b μ μ b Υ ˜ ( ϰ ) d ϰ F 2 π e 1 + 4 π 2 + 4 [ Υ ˜ ( μ ) Υ ˜ ( b ) ] .
If one lays α 1 , and Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then we get (see [78])
e 2 Υ ˜ ( μ + b 2 ) F 1 b μ μ b Υ ˜ ( ϰ ) d ϰ F 2 π e 1 + 4 π 2 + 4 [ Υ ˜ ( μ ) Υ ˜ ( b ) ] .
Let α 1 and Υ * ( ϰ ,   θ ) Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 6, we acquire (see [78])
e 2 Υ ( μ + b 2 ) 1 b μ μ b Υ ( ϰ ) d ϰ 2 π e 1 + 4 π 2 + 4 [ Υ ( μ ) + Υ ( b ) ] .
If Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) and θ = 1 , then, from Theorem 6, we have
e 2 Υ ( μ + b 2 ) 1 α 2 ( 1 e ν ) [ μ + α   Υ ( b ) + b α   Υ ( μ ) ] ν 1 e ν S ( ν ) Υ ( μ ) + Υ ( b ) 2 .
Let α 1 and Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 6, we achieve (see [73])
e 2 Υ ( μ + b 2 ) 1 b μ μ b Υ ( ϰ ) d ϰ 2 π e 1 + 4 π 2 + 4 [ Υ ( μ ) + Υ ( b ) ] .
Example 1.
Let  α = 1 2 ,   ϰ [ 0 , 1 ] , and the  F N V M   Υ ˜ : [ μ ,   b ] = [ 0 ,   1 ] R C ,  defined by
Υ ˜ ( ϰ ) ( θ ) = { θ 2 ϰ 2 3 2 ϰ 2 θ [ 2 ϰ 2 ,   3 ]   2 ( 1 + e x ) θ 2 e x 1 θ ( 3 ,   2 ( 1 + e x ) ]   0 o t h e r w i s e ,  
then, for each  θ [ 0 ,   1 ] ,  we have  Υ θ ( ϰ ) = [ 2 ( 1 θ ) ϰ 2 + 3 θ , 2 ( 1 θ ) ( 1 + e x ) + 3 θ ] . Since, for each  θ [ 0 ,   1 ] , the left and right end point functions  Υ * ( ϰ , θ ) = 2 ( 1 θ ) ϰ 2 + 3 θ ,   Υ * ( ϰ ,   θ ) = 2 ( 1 θ ) ( 1 + e x ) + 3 θ  are exponential trigonometric convex and concave functions, respectively, then  Υ ˜ ( ϰ )  is the  U D E T - convex  F N V M . We readily see that  Υ ˜ L ( [ μ ,   b ] , R C )  and  ν = 1 α α ( b μ ) = 1 . Now, we calculate the following:
e 2 Υ * ( μ + b 2 ,   θ ) = e 2 Υ * ( 1 2 ,   θ ) = e 2 2 ( 1 θ ) + 3 e 2 θ
e 2 Υ * ( μ + b 2 ,   θ ) = Υ * ( 5 2 ,   θ ) = 2 ( 1 θ ) 1 + e 2 + 3 e 2 θ
ν 1 e ν S ( ν ) Υ * ( μ , θ ) + Υ * ( b ,   θ ) 2 = 8 + 2 π e 2 ( 16 + π 2 ) ( 1 e 1 ) ( 1 + 2 θ )
ν 1 e ν S ( ν ) Υ * ( μ , θ ) + Υ * ( b ,   θ ) 2 = 8 + 2 π e 2 ( 16 + π 2 ) ( 1 e 1 ) ( ( 1 θ ) ( 3 + e ) + 3 θ ) .
Note that
1 α 2 ( 1 e ν ) [ μ + α   Υ * ( b ,   θ ) + b α   Υ * ( μ , θ ) ]   = 1 2 ( 1 e 1 ) 0 1 e x   . ( 2 ( 1 θ ) ϰ 2 + 3 θ ) d ϰ   + 1 2 ( 1 e 1 ) 0 1 e x   . ( 2 ( 1 θ ) ϰ 2 + 3 θ ) d ϰ   = ( 1 θ ) 1 1 e 1 [ 1 2 e 1 + 2 5 e 1 ] + 3 2 θ   = ( 1 θ ) 3 7 e 1 1 e 1 3 2 θ .
1 α 2 ( 1 e ν ) [ μ + α   Υ * ( b ,   θ ) + b α   Υ * ( μ , θ ) ]   = 1 2 ( 1 e 1 ) 0 1 e ( 1 ϰ )   .   ( 2 ( 1 θ ) ( 1 + e x ) + 3 θ ) d ϰ   + 1 2 ( 1 e 1 ) 0 1 e x   .   ( 2 ( 1 θ ) ( 1 + e x ) + 3 θ ) d ϰ   = ( 1 θ ) 1 1 e 1 [ e 2 3 e 1 2 + 3 e 1 ] + 3 2 θ   = ( 1 θ ) e 5 e 1 + 6 2 ( 1 e 1 ) + 3 2 θ .
Therefore,
[ e 2 2 ( 1 θ ) + 3 e 2 θ , 2 ( 1 θ ) 1 + e 2 + 3 e 2 ] I [ ( 1 θ ) 3 7 e 1 1 e 1 + 3 2 θ , ( 1 θ ) e 5 e 1 + 6 2 ( 1 e 1 ) + 3 2 θ ] I [ 8 + 2 π e 2 ( 16 + π 2 ) ( 1 e 1 ) ( 1 + 2 θ ) , 8 + 2 π e 2 ( 16 + π 2 ) ( 1 e 1 ) ( ( 1 θ ) ( 3 + e ) + 3 θ ) ] ,
and Theorem 6 is verified.
Here are midpoint H H - type inequalities for U D E T - convex F N V M .
Theorem 7.
Let  Υ ˜ : [ μ ,   b ] R C  be an  U D E T - convex  F N V M  on  [ μ ,   b ] ,  with   θ  levels defined by the family of i-v-ms  Υ θ : [ μ ,   b ] X C +  given by  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ]  for all  ϰ [ μ ,   b ]  and for all  θ [ 0 ,   1 ] . If  Υ ˜ L ( [ μ ,   b ] , R C ) ; then,
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) ] F ν 2 ( 1 e ν 2 ) P ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .  
If Υ ˜ ( ϰ ) is U D E T concave F N V M , then
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) ] F ν 1 e ν 2 P ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
where
P ( ν ) = π ( 4 e 1 2 3 2 e ν + 1 2 ) 2 5 2 ν e ν + 1 2 + 2 5 2 e ν + 1 2 4 ν 2 8 ν + π 2 + 4 + 8 ν + 2 3 2 π e ν + 1 2 2 5 2 ( ν + 1 ) e ν + 1 2 + 8 4 ν 2 + 8 ν + π 2 + 4 ,   ν = 1 α α ( b μ ) ,   and   1 > α > 0 .
Proof. 
Let Υ ˜ : [ μ ,   b ] R C be an U D E T - convex F N V M . Then, by hypothesis, we have
Υ ˜ ( μ + b 2 ) F sin π 4 e Υ ˜ ( 𝓉 2 μ + 2 𝓉 2 b ) cos π 4 e Υ ˜ ( 2 𝓉 2 μ + 𝓉 2 b ) .
After simplification, we get that
2 Υ ˜ ( μ + b 2 ) F 2 e [ Υ ˜ ( 𝓉 2 μ + 2 𝓉 2 b ) Υ ˜ ( 2 𝓉 2 μ + 𝓉 2 b ) ] .
Therefore, for every θ [ 0 ,   1 ] , we have
2 Υ * ( μ + b 2 ,   θ ) 2 e [ Υ * ( 𝓉 2 μ + 2 𝓉 2 b ,   θ ) + Υ * ( 2 𝓉 2 μ + 𝓉 2 b ,   θ ) ] , 2 Υ * ( μ + b 2 ,   θ ) 2 e [ Υ * ( 𝓉 2 μ + 2 𝓉 2 b ,   θ ) + Υ * ( 2 𝓉 2 μ + 𝓉 2 b ,   θ ) ] .
Taking Υ * ( . ,   θ ) and multiplying both sides by e ν 𝓉 2 and integrating the obtained results with respect to 𝓉 from 0 to 1 , we have
2 0 1 e ν 𝓉 2 Υ * ( μ + b 2 ,   θ ) d 𝓉 2 e [ 0 1 e ν 𝓉 2 Υ * ( 𝓉 2 μ + 2 𝓉 2 b , θ ) d 𝓉 + 0 1 e ν 𝓉 2 Υ * ( 2 𝓉 2 μ + 𝓉 2 b ,   θ ) d 𝓉 ] .
Let u = 𝓉 2 μ + 2 𝓉 2 b and x = 2 𝓉 2 μ + 𝓉 2 b . Then, we have
2 0 1 e ν 𝓉 2 Υ * ( μ + b 2 ,   θ ) d 𝓉   2 e 1 b μ   μ + b 2 b e ( 1 α α ( b u ) ) Υ * ( u , θ ) d u + 1 b μ μ + b 2 b e ( 1 α α ( x μ ) ) Υ * ( ϰ , θ ) d ϰ   = 2 e α b μ I μ + b 2 + α   Υ b ,   θ + I μ + b 2 α   Υ μ ,   θ .
Now, taking the right side of Equation (64), we have
0 1 e ν 𝓉 2 Υ * ( μ + b 2 ,   θ ) d 𝓉 = 2 ( 1 e ν ) ν Υ * ( μ + b 2 ,   θ ) .
From (64) and (65), we have
4 · 1 e ν ν Υ * ( μ + b 2 ,   θ ) 2 e · 2 ( 1 μ ) b μ [ ( μ + b 2 ) + α   Υ * ( b ,   θ ) + ( μ + b 2 ) α   Υ * ( μ ,   θ ) ] .
Similarly, for Υ * ( ϰ , θ ) , we have
4 · 1 e ν ν Υ * ( μ + b 2 ,   θ ) 2 e · 2 ( 1 μ ) b μ [ ( μ + b 2 ) + α   Υ * ( b ,   θ ) + ( μ + b 2 ) α   Υ * ( μ ,   θ ) ] .
From (66) and (67), we have
2 · 1 e ν ν [ Υ * ( μ + b 2 ,   θ ) ,   Υ * ( μ + b 2 ,   θ ) ] I 2 e · 1 μ b μ [ ( μ + b 2 ) + α   Υ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) ,   ( μ + b 2 ) + α   Υ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) ] .
That is
2 · 1 e ν ν   Υ θ ( μ + b 2 ) I 2 e · 1 μ b μ [ ( μ + b 2 ) + α   Υ θ ( b ) + ( μ + b 2 ) α   Υ θ ( μ ) ] .
For the right side of Equation (62), since Υ ˜ is an U D E T - convex F N V M , then we deduce that
Υ ˜ ( 𝓉 2 μ + 2 𝓉 2 b ) F sin π 𝓉 4 e 2 𝓉 2 Υ ˜ ( μ ) cos π 𝓉 4 e 𝓉 2 Υ ˜ ( b ) ,
and
Υ ˜ ( 2 𝓉 2 μ + 𝓉 2 b ) F cos π 𝓉 4 e 𝓉 2 Υ ˜ ( μ ) sin π 𝓉 4 e 2 𝓉 2 Υ ˜ ( b ) .
Adding (69) and (70), we have
Υ ˜ ( 𝓉 2 μ + 2 𝓉 2 b ) Υ ˜ ( 2 𝓉 2 μ + 𝓉 2 b ) F [ Υ ˜ ( μ ) Υ ˜ ( b ) ] [ sin π 𝓉 4 e 2 𝓉 2 + cos π 𝓉 4 e 𝓉 2 ] .
Since Υ ˜ is F N V M , then, for each   θ [ 0 ,   1 ] , we have
Υ 𝓉 2 μ + 2 𝓉 2 b ,   θ + Υ 2 𝓉 2 μ + 𝓉 2 b ,   θ Υ μ ,   θ + Υ b ,   θ s i n π 𝓉 4 e 2 𝓉 2 + c o s π 𝓉 4 e 𝓉 2 ,   Υ 𝓉 2 μ + 2 𝓉 2 b ,   θ + Υ 2 𝓉 2 μ + 𝓉 2 b ,   θ Υ μ ,   θ + Υ b ,   θ s i n π 𝓉 4 e 2 𝓉 2 + c o s π 𝓉 4 e 𝓉 2 .  
Taking Υ * ( . ,   θ ) from (72) and multiplying the inequality with e ν 𝓉 2 and integrating the results with 𝓉 from 0 to 1 , we have
0 1 e e ν 𝓉 2 Υ * ( 𝓉 2 μ + 2 𝓉 2 b ,   θ ) d 𝓉 + 0 1 e e ν 𝓉 2 Υ * ( 2 𝓉 2 μ + 𝓉 2 b   ,   θ ) d 𝓉   [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] 0 1 e e ν 𝓉 2 [ sin π 𝓉 4 2 𝓉 2 + cos π 𝓉 4 e 𝓉 2 ] d 𝓉 ,   = ( π ( 4 e 1 2 3 2 e ν + 1 2 ) 2 5 2 ν e ν + 1 2 + 2 5 2 e ν + 1 2 4 ν 2 8 ν + π 2 + 4 + 8 ν + 2 3 2 π e ν + 1 2 2 5 2 ( ν + 1 ) e ν + 1 2 + 8 4 ν 2 + 8 ν + π 2 + 4 ) [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] .
In a manner identical to that described previously, we have Υ * ( . ,   θ )
0 1 e e ν 𝓉 2 Υ * ( 𝓉 2 μ + 2 𝓉 2 b ,   θ ) d 𝓉 + 0 1 e e ν 𝓉 2 Υ * ( 2 𝓉 2 μ + 𝓉 2 b ,   θ ) d 𝓉   [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] 0 1 e ν 𝓉 [ sin π 𝓉 4 e 2 𝓉 2 + cos π 𝓉 4 e 𝓉 2 ] d 𝓉 ,   = ( π ( 4 e 1 2 3 2 e ν + 1 2 ) 2 5 2 ν e ν + 1 2 + 2 5 2 e ν + 1 2 4 ν 2 8 ν + π 2 + 4 + 8 ν + 2 3 2 π e ν + 1 2 2 5 2 ( ν + 1 ) e ν + 1 2 + 8 4 ν 2 + 8 ν + π 2 + 4 ) [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] .
From (73) and (74), we have
0 1 e ν 𝓉 Υ θ ( 𝓉 2 μ + 2 𝓉 2 b   ) d 𝓉 + 0 1 e ν 𝓉 Υ θ ( 2 𝓉 2 μ + 𝓉 2 b   ) d 𝓉 I ( π ( 4 e 1 2 3 2 e ν + 1 2 ) 2 5 2 ν e ν + 1 2 + 2 5 2 e ν + 1 2 4 ν 2 8 ν + π 2 + 4 + 8 ν + 2 3 2 π e ν + 1 2 2 5 2 ( ν + 1 ) e ν + 1 2 + 8 4 ν 2 + 8 ν + π 2 + 4 ) [ Υ θ ( μ ) + Υ θ ( b ) ] .
From (68) and (75), we have
e 2 Υ θ ( μ + b 2 ) I 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α Υ θ ( b ) + ( μ + b 2 ) α Υ θ ( μ ) ] I ν 2 ( 1 e ν 2 ) P ( ν ) Υ θ ( μ ) + Υ θ ( b ) 2 .
that is
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) ] F ν 2 ( 1 e ν 2 ) P ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
Hence, the required results. □
Remark 5.
From Theorem 7, we see that
If one lays Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then one acquires the following outcome (see [78]):
e 2 Υ ˜ ( μ + b 2 ) F 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) ] F ν 2 ( 1 e ν 2 ) P ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
Let Υ ϰ , θ Υ ϰ , θ and θ = 1 , . Then, from Theorem 7, we have (see [78])
e 2 Υ ( μ + b 2 ) 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ ( b ) + ( μ + b 2 ) α   Υ ( μ ) ] ν 2 ( 1 e ν 2 ) P ( ν ) Υ ( μ ) + Υ ( b ) 2 .
If α 1 , that is
lim α 1 ν = lim α 1 1 α α ( b μ ) = 0 ,
then
lim α 1 ν 1 e ν 2 ( π ( 4 e 1 2 3 2 e ν + 1 2 ) 2 5 2 ν e ν + 1 2 + 2 5 2 e ν + 1 2 4 ν 2 8 ν + π 2 + 4 + 8 ν + 2 3 2 π e ν + 1 2 2 5 2 ( ν + 1 ) e ν + 1 2 + 8 4 ν 2 + 8 ν + π 2 + 4 ) = 4 ( 2 π + 4 e ) e ( π 2 + 4 ) lim α 1 1 α 2 ( 1 e e ν 2 ) = 1 b μ .
The next finding, which is likewise novel, is as follows:
e 2 Υ ˜ ( μ + b 2 ) F 1 b μ μ b Υ ˜ ( ϰ ) d ϰ F 2 π + 4 e e ( π 2 + 4 ) [ Υ ˜ ( μ ) Υ ˜ ( b ) ] .
If one lays α 1 , and Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then we have (see [78])
e 2 Υ ˜ ( μ + b 2 ) F 1 b μ μ b Υ ˜ ( ϰ ) d ϰ F 2 π + 4 e e ( π 2 + 4 ) [ Υ ˜ ( μ ) Υ ˜ ( b ) ] .
Let α 1 and Υ * ( ϰ ,   θ ) Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 7, we have (see [78])
e 2 Υ ( μ + b 2 ) 1 b μ μ b Υ ( ϰ ) d ϰ 2 π + 4 e e ( π 2 + 4 ) [ Υ ( μ ) + Υ ( b ) ] .
If Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) and θ = 1 , then, from Theorem 7, we have
e 2 Υ ( μ + b 2 ) 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ ( b ) + ( μ + b 2 ) α   Υ ( μ ) ] ν 2 ( 1 e ν 2 ) P ( ν ) Υ ( μ ) + Υ ( b ) 2 .
Let α 1 and Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 7, we have (see [73])
e 2 Υ ( μ + b 2 ) 1 b μ μ b Υ ( ϰ ) d ϰ 2 π + 4 e e ( π 2 + 4 ) [ Υ ( μ ) + Υ ( b ) ] .
Here are some results on H H - type inequalities for the products of U D E T - convex FNVM, which are known as Pachpatte-type inequalities.
Theorem 8.
Let  Υ ˜ , Τ ˜   : [ μ ,   b ] R C  be two  U D E T convex  F N V M s on  [ μ ,   b ] ,  with  θ  levels  Υ θ ,   Τ θ : [ μ ,   b ] X C +  defined by  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ]  and  Τ θ ( ϰ ) = [ Τ * ( ϰ , θ ) ,   Τ * ( ϰ , θ ) ]  for all  ϰ [ μ ,   b ]  and for all  θ [ 0 ,   1 ] . If  Υ ˜ Τ ˜ L ( [ μ ,   b ] , R C ) , then
α b μ [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α Υ ˜ ( μ ) Τ ˜ ( μ ) ] F A ( ν ) Δ ( μ , b ) π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( μ , b ) .  
If Υ ˜ ( ϰ ) is U D E T - concave FNVM, then
α b μ [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α Υ ˜ ( μ ) Τ ˜ ( μ ) ] F A ( ν ) Δ ( μ , b ) π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( μ , b ) .
where
A ν = e ν 2 8 e 2 8 ν e 2 + π 2 e 2 + 2 ν 2 e 2 π 2 e ν 2 ν 2 ν 2 4 ν + π 2 + 4 + 8 ν π 2 e ν 2 + π 2 + 2 ν 2 + 8 2 ν + 2 ν 2 + 4 ν + π 2 + 4 ,   ν = 1 α α b μ ,   1 > α > 0 , Δ μ , b = Υ ~ μ Τ ~ μ     Υ ~ b Τ ~ b , μ , b = Υ ~ μ Τ ~ b Υ ~ b Τ ~ μ ,
and
Δ θ ( μ , b ) = [ Δ * ( ( μ , b ) ,   θ ) ,   Δ * ( ( μ , b ) ,   θ ) ]
and
θ ( μ , b ) = [ * ( ( μ , b ) ,   θ ) ,   * ( ( μ , b ) ,   θ ) ] .
Proof. 
Since Υ ˜ ,   Τ ˜ both are U D E T - convex FNVMs, for each θ [ 0 ,   1 ] ,taking the left end points functions, we have
    Υ ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) sin π 𝓉 2 e 1 𝓉 Υ ( μ ,   θ ) + cos π 𝓉 2 e 𝓉 Υ ( b ,   θ ) .
and
    Τ ( 𝓉 μ + ( 1 𝓉 ) b ,   θ ) t sin π 𝓉 2 e 1 𝓉 Τ ( μ ,   θ ) + cos π 𝓉 2 e 𝓉 Τ ( b ,   θ ) .  
From the definition of U D E T - convex F N V M s, it follows that 0 ˜ F Υ ˜ ( ϰ ) and 0 ˜ F Τ ˜ ( ϰ ) , so
Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ   s i n π 𝓉 2 e 1 𝓉 Υ μ , θ + c o s π 𝓉 2 e 𝓉 Υ b ,   θ s i n π 𝓉 2 e 1 𝓉 Τ μ , θ + c o s π 𝓉 2 e 𝓉 Τ b ,   θ   = s i n π 𝓉 2 e 1 𝓉 2 Υ μ , θ × Τ μ , θ + c o s π 𝓉 2 e 𝓉 2 Υ b ,   θ × Τ b ,   θ                   + c o s π 𝓉 2 s i n π 𝓉 2 e Υ μ , θ × Τ b ,   θ + c o s π 𝓉 2 s i n π 𝓉 2 e Υ b ,   θ × Τ μ , θ .
Analogously, we have
Υ 1 𝓉 μ + 𝓉 b ,   θ Τ 1 𝓉 μ + 𝓉 b ,   θ c o s π 𝓉 2 e 𝓉 2 Υ μ , θ × Τ μ , θ + s i n π 𝓉 2 e 1 𝓉 2 Υ b ,   θ × Τ b ,   θ                 + c o s π 𝓉 2 s i n π 𝓉 2 e Υ μ , θ × Τ b ,   θ + c o s π 𝓉 2 s i n π 𝓉 2 e Υ b ,   θ × Τ μ , θ .
Adding (85) and (86), we have
Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ           + Υ 1 𝓉 μ + 𝓉 b ,   θ × Τ 1 𝓉 μ + 𝓉 b ,   θ                             s i n π 𝓉 2 e 1 𝓉 2 + s i n π 𝓉 2 e 1 𝓉 2 Υ μ , θ × Τ μ , θ + Υ b ,   θ × Τ b ,   θ + 2 c o s π 𝓉 2 s i n π 𝓉 2 e Υ b ,   θ × Τ μ , θ + Υ μ , θ × Τ b ,   θ .
Taking the multiplication of (87) by e ν 𝓉 and integrating the obtained result with respect to 𝓉 over (0,1), we have
0 1 e ν 𝓉 Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ + e ν 𝓉 Υ 1 𝓉 μ + 𝓉 b ,   θ × Τ 1 𝓉 μ + 𝓉 b ,   θ d t                       Δ μ , b ,   θ 0 1 e ν 𝓉 s i n π 𝓉 2 e 1 𝓉 2 + s i n π 𝓉 2 e 1 𝓉 2 d t + 2 μ , b ,   θ 0 1 e ν 𝓉 c o s π 𝓉 2 s i n π 𝓉 2 e d 𝓉 .
It follows that
α b μ I μ + α   Υ b ,   θ × Τ b ,   θ + I b α   Υ μ , θ × Τ μ , θ             A ν Δ μ , b ,   θ + π e ν 1 e ν + 1 ν 2 + π 2 μ , b ,   θ .  
Similarly, for Υ * ( ϰ , θ ) , we have
α b μ [ μ + α   Υ * ( b ,   θ ) × Τ * ( b ,   θ ) + b α   Υ * ( μ , θ ) × Τ * ( μ , θ ) ] A ( ν ) Δ * ( ( μ , b ) ,   θ ) + π e ν 1 ( e ν + 1 ) ν 2 + π 2 * ( ( μ , b ) ,   θ ) ,
where
A ( ν ) = 0 1 e ν 𝓉 [ ( sin π 𝓉 2 e 1 𝓉 ) 2 + ( sin π 𝓉 2 e 1 𝓉 ) 2 ] d 𝓉 = e ν 2 ( 8 e 2 8 ν e 2 + π 2 e 2 + 2 ν 2 e 2 π 2 e ν ) 2 ( ν 2 ) ( ν 2 4 ν + π 2 + 4 ) + 8 ν π 2 e ν 2 + π 2 + 2 ν 2 + 8 2 ( ν + 2 ) ( ν 2 + 4 ν + π 2 + 4 ) ,
and
0 1 e ν 𝓉 cos π 𝓉 2   sin π 𝓉 2 e d 𝓉 = π e ν 1 ( e ν + 1 ) ν 2 + π 2 .
From (88) and (89), we have
α b μ [ μ + α   Υ * ( b ,   θ ) × Τ * ( b ,   θ ) + b α   Υ * ( μ , θ ) × Τ * ( μ , θ ) ,     μ + α   Υ * ( b ,   θ ) × Τ * ( b ,   θ ) + b α   Υ * ( μ , θ ) × Τ * ( μ ,   θ ) ] I A ( ν ) [ Δ * ( ( μ , b ) ,   θ ) ,   Δ * ( ( μ , b ) ,   θ ) ] + π e ν 1 ( e ν + 1 ) ν 2 + π 2 [ * ( ( μ , b ) ,   θ ) ,   * ( ( μ , b ) ,   θ ) ] .
That is
α b μ [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α Υ ˜ ( μ ) Τ ˜ ( μ ) ] F A ( ν ) Δ ( μ , b ) π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( μ , b ) .
and the theorem has been established. □
Remark 6.
We readily see from Theorem 8 that
If one lays Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then one obtains (see [78])
α b μ [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α Υ ˜ ( μ ) Τ ˜ ( μ ) ] F A ( ν ) Δ ( μ , b ) π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( μ , b ) .
Let Υ ϰ , θ Υ ϰ , θ and θ = 1 , . Then, from Theorem 8, we have (see [86])
α b μ [ μ + α   Υ ( b ) × Τ ( b ) + b α Υ ( μ ) × Τ ( μ ) ] A ( ν ) Δ ( μ , b ) + π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( μ , b ) .
If α 1 , that is
lim α 1 ν = lim α 1 1 α α ( b μ ) = 0 ,
then
lim α 1 ( e ν 2 ( 8 e 2 8 ν e 2 + π 2 e 2 + 2 ν 2 e 2 π 2 e ν ) 2 ( ν 2 ) ( ν 2 4 ν + π 2 + 4 ) + 8 ν π 2 e ν 2 + π 2 + 2 ν 2 + 8 2 ( ν + 2 ) ( π 2 + 4 ) ) = π 2 π 2 e 2 + 8 2 ( π 2 + 4 ) ,   lim α 1 π e ν 1 ( e ν + 1 ) ν 2 + π 2 = 2 π e .
The next finding, which is likewise novel, is as follows:
1 b μ μ b Υ ˜ ( x ) Τ ˜ ( x ) d ϰ F π 2 π 2 e 2 + 8 4 ( π 2 + 4 ) Δ ( μ , b ) 2 π e ( μ , b ) .
If one lays α 1 and Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then one can get (see [78])
1 b μ μ b Υ ˜ ( x ) Τ ˜ ( x ) d ϰ F π 2 π 2 e 2 + 8 4 ( π 2 + 4 ) Δ ( μ , b ) 2 π e ( μ , b ) .
Let α 1 and Υ * ( ϰ ,   θ ) Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 8, we have (see [86])
1 b μ μ b Υ ( x ) × Τ ( x ) d ϰ π 2 π 2 e 2 + 8 4 ( π 2 + 4 ) Δ ( μ , b ) + 2 π e ( μ , b ) .
If Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) and θ = 1 , then, from Theorem 8, we achieve
α b μ [ μ + α   Υ ( b ) × Τ ( b ) + b α Υ ( μ ) × Τ ( μ ) ] A ( ν ) Δ ( μ , b ) + π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( μ , b ) .
Let α 1 and Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 8, we have (see [73])
1 b μ μ b Υ ( x ) × Τ ( x ) d ϰ π 2 π 2 e 2 + 8 4 ( π 2 + 4 ) Δ ( μ , b ) + 2 π e ( μ , b ) .
Example 2.
Let  [ μ , b ] = [ 0 , 1 ] , α = 1 4  and
Υ ˜ ( x ) ( θ ) = { θ x 2 3 2 x 2 θ [ x 2 , 3 2   ] ,   x + 1 θ x 1 2 θ ( 3 2 ,   x + 1 ] ,   0 o t h e r w i s e ,  
Τ ˜ ( x ) ( θ ) = { θ 2 x 3 3 2 x 3               θ [ 2 x 3 ,   3 ] , 1 + e x θ   e x 2       θ ( 3 ,   1 + e x ] ,   0                       o t h e r w i s e .
Then, for each θ [ 0 ,   1 ] , we have Υ θ ( x ) = [ ( 1 θ ) x 2 + 3 2 θ , ( 1 θ ) ( x + 1 ) + 3 2 θ ] and T θ ( x ) = [ 2 ( 1 θ ) x 3 + 3 θ , ( 1 θ ) ( 1 + e x ) + 3 θ ] . It can be easily seen that, for each θ [ 0 ,   1 ] , the left and right end point functions Υ * ( x , θ ) = ( 1 θ ) x 2 + 3 2 θ ,   Υ * ( x ,   θ ) = ( 1 θ ) ( x + 1 ) + 3 2 θ , T * ( x , θ ) = 2 ( 1 θ ) x 3 + 3 θ , and T * ( x ,   θ ) = ( 1 θ ) ( 1 + e x ) + 3 θ are exponential trigonometric convex and concave functions; then, Υ ( x ) and T ( x ) both are U D E T convex F N V M s. We readily see that Υ ˜ ( x ) Τ ˜ ( x ) L ( [ μ ,   b ] , R C ) and ν = 1 α α ( b μ ) = 3 .
α b μ I μ + α   Υ b ,   θ × T b ,   θ + I b α   Υ μ ,   θ × T μ ,   θ = 0 1 e 3 1 x 1 θ x 2 + 3 2 θ 2 1 θ x 3 + 3 θ d x + 0 1 e 3 x 1 θ x 2 + 3 2 θ 2 1 θ x 3 + 3 θ d x = 1020 θ 2 + 174 θ + 264 e 3 1920 θ 2 + 3246 θ 2784 486 e 3
α b μ I μ + α   Υ b ,   θ × T b ,   θ + I b α   Υ μ ,   θ × T μ ,   θ = 0 1 e 3 1 x 1 θ x + 1 + 3 2 θ 1 θ 1 + e x + 3 θ d x + 0 1 e 3 x 1 θ x + 1 + 3 2 θ 1 θ 1 + e x + 3 θ d x = 9 e 4 θ 1 θ 7 + 4 e 3 8 θ 2 31 θ + 23 36 e 2 θ 2 7 θ + 5 + 133 θ 2 80 θ 37 144 e 3
Note that
A ( ν ) Δ * ( ( μ , b ) ,   θ ) = ( e 5 ( 2 e 2 + π 2 e 2 π 2 e 3 ) 2 ( 1 + π 2 ) + 50 π 2 e 5 + π 2 10 ( 25 + π 2 ) ) [ Υ * ( 0 ,   θ ) × T * ( 0 ,   θ ) + Υ * ( 1 ,   θ ) × T * ( 1 ) ] = ( 50 π 2 e 5 + π 2 10 ( 25 + π 2 ) e 5 ( 2 e 2 + π 2 e 2 π 2 e 3 ) 2 ( 1 + π 2 ) ) ( 5 θ 2 + 2 θ + 2 ) ,
A ( ν ) Δ * ( ( μ , b ) ,   θ ) = ( e 5 ( 2 e 2 + π 2 e 2 π 2 e 3 ) 2 ( 1 + π 2 ) + 50 π 2 e 5 + π 2 10 ( 25 + π 2 ) ) [ Υ * ( 0 ,   θ ) × T * ( 0 ,   θ ) + Υ * ( 1 ,   θ ) × T * ( 1 ,   θ ) ] = ( 50 π 2 e 5 + π 2 10 ( 25 + π 2 ) e 5 ( 2 e 2 + π 2 e 2 π 2 e 3 ) 2 ( 1 + π 2 ) ) ( e θ 2 θ 2 5 e θ + 11 θ + 4 e + 8 ) ,
π e ν 1 ( e ν + 1 ) ν 2 + π 2 * ( ( μ , b ) ,   θ ) = π e ν 1 ( e ν + 1 ) ν 2 + π 2 [ Υ * ( 0 ,   θ ) × T * ( 1 ,   θ ) + Υ * ( 1 ,   θ ) × T * ( 0 ,   θ ) ] = 3 θ ( θ + 2 ) ,
π e ν 1 ( e ν + 1 ) ν 2 + π 2 * ( ( μ , b ) ,   θ ) = π e ν 1 ( e ν + 1 ) ν 2 + π 2 [ Υ * ( 1 ,   θ ) × T * ( 1 ,   θ ) + Υ * ( 1 ,   θ ) × T * ( 0 ,   θ ) ] = 1 2 π e ν 1 ( e ν + 1 ) ν 2 + π 2 ( θ + 2 ) ( 5 + e + θ θ e ) .
Therefore, we have
A ( ν ) Δ θ ( μ , b ) + π e ν 1 ( e ν + 1 ) ν 2 + π 2 θ ( μ , b ) = ( 50 π 2 e 5 + π 2 10 ( 25 + π 2 ) e 5 ( 2 e 2 + π 2 e 2 π 2 e 3 ) 2 ( 1 + π 2 ) ) [ 5 θ 2 + 2 θ + 2 , e θ 2 θ 2 5 e θ + 11 θ + 4 e + 8 ] + π e ν 1 ( e ν + 1 ) ν 2 + π 2 [ 3 θ ( θ + 2 ) , 1 2 ( θ + 2 ) ( 5 + e + θ θ e ) ]
It follows that
[ ( 1020 θ 2 + 174 θ + 264 ) e 3 1920 θ 2 + 3246 θ 2784 486 e 3 , 9 e 4 ( θ 1 ) ( θ 7 ) + 4 e 3 ( 8 θ 2 31 θ + 23 ) 36 e ( 2 θ 2 7 θ + 5 ) + 133 θ 2 80 θ 37 144 e 3 ] I ( 50 π 2 e 5 + π 2 10 ( 25 + π 2 ) e 5 ( 2 e 2 + π 2 e 2 π 2 e 3 ) 2 ( 1 + π 2 ) ) [ 5 θ 2 + 2 θ + 2 , e θ 2 θ 2 5 e θ + 11 θ + 4 e + 8 ] + π e ν 1 ( e ν + 1 ) ν 2 + π 2 [ 3 θ ( θ + 2 ) , 1 2 ( θ + 2 ) ( 5 + e + θ θ e ) ]
and Theorem 8 has been validated.
Theorem 9.
Let  Υ ˜ , Τ ˜   : [ μ ,   b ] R C  be two  U D E T - convex  F N V M s, with  θ  levels defined by the family of i-v-ms given by  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ]  and  Τ θ ( ϰ ) = [ Τ * ( ϰ , θ ) ,   Τ * ( ϰ , θ ) ]  for all  ϰ [ μ ,   b ]  and for all  θ [ 0 ,   1 ] . If  Υ ˜ Τ L ( [ μ ,   b ] , R C ) , then
2 Υ ˜ ( μ + b 2 ) Τ ˜ ( μ + b 2 ) F 1 α e ( 1 e ν ) [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α   Υ ˜ ( μ ) Τ ˜ ( μ ) ] + ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) ( μ , b ) ν e ( 1 e ν ) A ( ν ) Δ ( μ , b ) .  
If Υ ˜ ( ϰ ) is U D E T - concave F N V M , then
2 Υ ˜ ( μ + b 2 ) Τ ˜ ( μ + b 2 ) F 1 α e ( 1 e ν ) [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α   Υ ˜ ( μ ) Τ ˜ ( μ ) ] + ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) ( μ , b ) ν e ( 1 e ν ) A ( ν ) Δ ( μ , b ) .
where
A ( ν ) = e ν 2 ( 8 e 2 8 ν e 2 + π 2 e 2 + 2 ν 2 e 2 π 2 e ν ) 2 ( ν 2 ) ( ν 2 4 ν + π 2 + 4 ) + 8 ν π 2 e ν 2 + π 2 + 2 ν 2 + 8 2 ( ν + 2 ) ( ν 2 + 4 ν + π 2 + 4 ) ,   ν = 1 α α ( b μ ) ,   1 > α > 0 ,
and
  Δ ( μ , b ) = Υ ˜ ( μ ) Τ ˜ ( μ ) Υ ˜ ( b ) Τ ˜ ( b ) ,   ( μ , b ) = Υ ˜ ( μ ) Τ ˜ ( b ) Υ ˜ ( b ) Τ ˜ ( μ ) ,
and
Δ θ ( μ , b ) = [ Δ * ( ( μ , b ) ,   θ ) ,   Δ * ( ( μ , b ) ,   θ ) ]
and
θ ( μ , b ) = [ * ( ( μ , b ) ,   θ ) ,   * ( ( μ , b ) ,   θ ) ] .
Proof. 
Consider Υ ˜ , Τ ˜   : [ μ ,   b ] R C are U D E T - convex F N V M s. Then, by hypothesis, for each θ [ 0 ,   1 ] , we have
Υ μ + b 2   , θ × Τ μ + b 2 , θ   1 2 e Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ + Υ 1 𝓉 μ + 1 𝓉 b ,   θ × Τ 1 𝓉 μ + 𝓉 b ,   θ         + 1 2 e Υ 1 𝓉 μ + 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ + Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 1 𝓉 μ + 𝓉 b ,   θ         1 2 e Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ + Υ 1 𝓉 μ + 𝓉 b ,   θ × Τ 1 𝓉 μ + 𝓉 b ,   θ       + 1 2 e c o s π 𝓉 2 e 𝓉 Υ μ ,   θ + s i n π 𝓉 2 e 1 𝓉 Υ b ,   θ × s i n π 𝓉 2 e 1 𝓉 Τ μ ,   θ + c o s π 𝓉 2 e 𝓉 Τ b ,   θ + s i n π 𝓉 2 e 1 𝓉 Υ μ ,   θ + c o s π 𝓉 2 e 𝓉 Υ b ,   θ × c o s π 𝓉 2 e 𝓉 Τ μ ,   θ + s i n π 𝓉 2 e 1 𝓉 Τ b ,   θ                                                         = 1 2 e Υ 𝓉 μ + 1 𝓉 b ,   θ × Τ 𝓉 μ + 1 𝓉 b ,   θ + Υ 1 𝓉 μ + 𝓉 b ,   θ × Τ 1 𝓉 μ + 𝓉 b ,   θ     + 1 2 e 2 c o s π 𝓉 2 s i n π 𝓉 2 e μ , b ,   θ + s i n π 𝓉 2 e 1 𝓉 2 + s i n π 𝓉 2 e 1 𝓉 2 Δ μ , b ,   θ .                          
Taking the multiplication of (101) with e ν 𝓉 and integrating over ( 0 ,   1 ) , we get
  0 1 e ν 𝓉 Υ μ + b 2 , θ × Τ μ + b 2 , θ d 𝓉               1 2 e   μ b e ν 𝓉 Υ ϰ , θ × Τ ϰ , θ d 𝓉 + μ b e ν 𝓉 Υ s , θ × Τ s , θ d 𝓉   + μ , b ,   θ 2 e 0 1 e ν 𝓉 2 c o s π 𝓉 2 s i n π 𝓉 2 e d t + Δ μ , b ,   θ 2 e 0 1 e ν 𝓉 s i n π 𝓉 2 e 1 𝓉 2 + s i n π 𝓉 2 e 1 𝓉 2 d t .       1 e ν ν   0 1 e ν 𝓉 Υ μ + b 2 , θ × Τ μ + b 2 , θ           α 2 e b μ I μ + α   Υ b ,   θ × Τ b ,   θ + I b α   Υ μ ,   θ × Τ μ ,   θ + π e ν 1 e ν + 1 2 e ν 2 + π 2 μ , b ,   θ 2 e + 1 2 e A ν Δ μ , b ,   θ 2 e .
Similarly, for , we have
1 e ν ν   Υ * ( μ + b 2 , θ ) × Τ * ( μ + b 2 , θ ) α 2 e ( b μ ) [ μ + α   Υ * ( b ,   θ ) × Τ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) × Τ * ( μ ,   θ ) ]   + π e ν 1 ( e ν + 1 ) 2 e ( ν 2 + π 2 ) * ( ( μ , b ) ,   θ ) 2 e + 1 2 e A ( ν ) Δ * ( ( μ , b ) ,   θ ) 2 e .
From (102) and (103), we have
2 [ Υ * ( μ + b 2 , θ ) × Τ * ( μ + b 2 , θ ) ,   Υ * ( μ + b 2 , θ ) × Τ * ( μ + b 2 , θ ) ] I 1 α e ( b μ ) [ μ + α   Υ * ( b ,   θ ) × Τ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) × Τ * ( μ ,   θ ) ,     μ + α   Υ * ( b ,   θ ) × Τ * ( b ,   θ ) + b α   Υ * ( μ ,   θ ) × Τ * ( μ ,   θ ) ] . + ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) [ * ( ( μ , b ) ,   θ ) ,   * ( ( μ , b ) ,   θ ) ] + ν e ( 1 e ν ) A ( ν ) [ * ( ( μ , b ) ,   θ ) ,   * ( ( μ , b ) ,   θ ) ] ,
where
A ( ν ) = e ν 2 ( 8 e 2 8 ν e 2 + π 2 e 2 + 2 ν 2 e 2 π 2 e ν ) 2 ( ν 2 ) ( ν 2 4 ν + π 2 + 4 ) + 8 ν π 2 e ν 2 + π 2 + 2 ν 2 + 8 2 ( ν + 2 ) ( ν 2 + 4 ν + π 2 + 4 ) .
That is
2 Υ ˜ ( μ + b 2 ) Τ ˜ ( μ + b 2 ) F α e ( b μ ) [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α   Υ ˜ ( μ ) Τ ˜ ( μ ) ] ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) ( μ , b ) ν e ( 1 e ν ) A ( ν ) Δ ( μ , b ) .
Hence, the required results. □
Remark 7.
We readily see from Theorem 9 that
If one lays Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then we have (see [78])
2 Υ ˜ ( μ + b 2 ) Τ ˜ ( μ + b 2 ) F α e ( b μ ) [ μ + α   Υ ˜ ( b ) Τ ˜ ( b ) b α Υ ˜ ( μ ) Τ ˜ ( μ ) ] ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) ( μ , b ) ν e ( 1 e ν ) A ( ν ) Δ ( μ , b ) .
Let Υ * ( ϰ , θ ) Υ * ( ϰ , θ ) and θ = 1 , . Then, we have (see [86])
2 Υ ( μ + b 2 ) × Τ ( μ + b 2 ) α e ( b μ ) [ μ + α   Υ ( b ) × Τ ( b ) + b α   Υ ( μ ) × Τ ( μ ) ] + ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) ( μ , b ) + ν e ( 1 e ν ) A ( ν ) Δ ( μ , b ) .
If α 1 , that is
lim α 1 ν = lim α 1 1 α α ( b μ ) = 0 ,
then
lim α 1 1 α e ( 1 e ν ) = 1 e ( b μ ) ,
lim α 1 ν e ( 1 e ν ) ( e ν 2 ( 8 e 2 8 ν e 2 + π 2 e 2 + 2 ν 2 e 2 π 2 e ν ) 2 ( ν 2 ) ( ν 2 4 ν + π 2 + 4 ) + 8 ν π 2 e ν 2 + π 2 + 2 ν 2 + 8 2 ( ν + 2 ) ( π 2 + 4 ) ) = π 2 π 2 e 2 + 8 2 e ( π 2 + 4 ) , lim α 1 ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) = 2 π e 2 .
The next finding, which is likewise novel, is as follows:
2 Υ ˜ ( μ + b 2 ) Τ ˜ ( μ + b 2 ) F 2 e ( b μ ) μ b Υ ˜ ( x ) Τ ˜ ( x ) d ϰ 2 π e 2 Δ ( μ , b ) π 2 π 2 e 2 + 8 2 e ( π 2 + 4 ) ( μ , b ) .
If one lays α 1 and Υ ˜ is a lower U D E T - convex F N V M on then we achieve (see [78])
2 Υ ˜ ( μ + b 2 ) Τ ˜ ( μ + b 2 ) F 2 e ( b μ ) μ b Υ ˜ ( x ) Τ ˜ ( x ) d ϰ 2 π e 2 Δ ( μ , b ) π 2 π 2 e 2 + 8 2 e ( π 2 + 4 ) ( μ , b ) .
Let α 1 and Υ * ( ϰ ,   θ ) Υ * ( ϰ ,   θ ) with . Then, from Theorem 9, we have (see [86])
2 Υ ( μ + b 2 ) × Τ ( μ + b 2 ) 2 e ( b μ ) μ b Υ ( x ) × Τ ( x ) d ϰ + 2 π e 2 Δ ( μ , b ) + π 2 π 2 e 2 + 8 2 e ( π 2 + 4 ) ( μ , b ) .
If Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) and then, from Theorem 9, we obtain
2 Υ ( μ + b 2 ) × Τ ( μ + b 2 ) α e ( b μ ) [ μ + α   Υ ( b ) × Τ ( b ) + b α   Υ ( μ ) × Τ ( μ ) ] + ν π e ν 1 ( e ν + 1 ) e ( 1 e ν ) ( ν 2 + π 2 ) ( μ , b ) + ν e ( 1 e ν ) A ( ν ) Δ ( μ , b ) .
Let α 1 and Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 9, we have (see [73])
2 Υ ( μ + b 2 ) × Τ ( μ + b 2 ) 2 e ( b μ ) μ b Υ ( x ) × Τ ( x ) d ϰ + 2 π e 2 Δ ( μ , b ) + π 2 π 2 e 2 + 8 2 e ( π 2 + 4 ) ( μ , b ) .
Here, we have presented some multiple inequalities in one result with the help of the midpoint and end points of the interval [ μ ,   b ] , and this outcome is very interesting because this is a generalization of some new and classical inequalities.
Theorem 10.
Let  Υ ˜ : [ μ ,   b ] R C  be an UDET-convex  F N V M  on  [ μ ,   b ] ,  with  θ  levels defined by the family of i-v-ms  Υ θ : [ μ ,   b ] X C +  given by  Υ θ ( ϰ ) = [ Υ * ( ϰ , θ ) ,   Υ * ( ϰ , θ ) ]  for all  ϰ [ μ ,   b ]  and for all  θ [ 0 ,   1 ] . If  Υ ˜ L ( [ μ ,   b ] , R C ) , then
e Υ ˜ ( μ + b 2 ) F e 2 [ Υ ˜ ( 3 μ + b 2 ) Υ ˜ ( μ + 3 b 2 ) ] F 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ ˜ ( μ + b 2 ) ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) b α   Υ ˜ ( μ + b 2 ) ] F ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ ˜ ( μ ) Υ ˜ ( b ) 2 Υ ˜ ( μ + b 2 ) ) F ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .  
If Υ ˜ ( ϰ ) is U D E T - concave F N V M , then
e Υ ˜ ( μ + b 2 ) F e 2 [ Υ ˜ ( 3 μ + b 4 ) Υ ˜ ( μ + 3 b 4 ) ] F 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ ˜ ( μ + b 2 ) ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) b α   Υ ˜ ( μ + b 2 ) ] F ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ ˜ ( μ ) Υ ˜ ( b ) 2 Υ ˜ ( μ + b 2 ) ) F ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
where
K ν = 2 ν + 2 π e ν + 2 2 + 4 ν 2 + 4 ν + π 2 + 4 + 2 π e 1 + 2 e ν 2 2 + ν ν 2 4 ν + π 2 + 4 ,   ν = 1 α α ( b μ ) ,   and   1 > α > 0 .
Proof. 
Take [ μ ,   μ + b 2 ] , we deduce that
Υ ~ 3 μ + b 4 F s i n π 4 e Υ ~ 𝓉 μ + 1 𝓉 μ + b 2 c o s π 4 e Υ ~ 1 𝓉 μ + 𝓉 μ + b 2 .
After simplification, we get that
2 Υ ˜ ( 3 μ + b 4 ) F 2 e [ Υ ˜ ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 ) Υ ˜ ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ) ] .
Therefore, for every θ [ 0 ,   1 ] , we have
2 Υ * ( 3 μ + b 4 ,   θ ) 2 e [ Υ * ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 ,   θ ) + Υ * ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ,   θ ) ] ,
2 Υ * ( 3 μ + b 4 ,   θ ) 2 e [ Υ * ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 ,   θ ) + Υ * ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ,   θ ) ] .
Taking Υ * ( . ,   θ ) and multiplying both sides by e ν 𝓉 2 and integrating the obtained results with respect to 𝓉 from 0 to 1 , we have
0 1 e ν 𝓉 2 Υ * ( 3 μ + b 4 ,   θ ) d 𝓉 2 e [ 0 1 e ν 𝓉 Υ * ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 , θ ) d 𝓉 + 0 1 e ν 𝓉 Υ * ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ,   θ ) d 𝓉 ] .  
Let u = 𝓉 μ + ( 1 𝓉 ) μ + b 2 and x = ( 1 𝓉 ) μ + 𝓉 μ + b 2 . Then, we have
0 1 e ν 𝓉 2 Υ * ( 3 μ + b 4 ,   θ ) d 𝓉   2 e 1 b μ   μ + b 2 b e ( 1 α α ( μ + b 2 u ) ) Υ * ( u , θ ) d u + 1 b μ μ + b 2 b e ( 1 α α ( x μ ) ) Υ * ( ϰ , θ ) d ϰ = 2 e α b μ [ μ + α   Υ * ( μ + b 2 ,   θ ) + ( μ + b 2 ) α   Υ * ( μ ,   θ ) ] .
Now, taking the right side of Equation (113), we have
0 1 e ν 𝓉 2 Υ * ( 3 μ + b 2 ,   θ ) d 𝓉 = 2 ( 1 e ν ) ν Υ * ( 3 μ + b 4 ,   θ ) .
From (113) and (114), we deduce that
2 ( 1 e ν ) ν Υ * ( 3 μ + b 4 ,   θ ) 2 e 1 α b μ [ μ + α   Υ * ( μ + b 2 ,   θ ) + ( μ + b 2 ) α   Υ * ( μ ,   θ ) ] .
Similarly, for Υ * ( . ,   θ ) from (115), we have
2 ( 1 e ν ) ν Υ * ( 3 μ + b 4 ,   θ ) 2 e 1 α b μ [ μ + α Υ * ( μ + b 2 ,   θ ) + ( μ + b 2 ) α Υ * ( μ ,   θ ) ] .
From (115) and (116), we deduce that
2 ( 1 e ν ) ν Υ θ ( 3 μ + b 4 ,   θ ) I 2 e 1 α b μ [ μ + α   Υ θ ( μ + b 2 ,   θ ) + ( μ + b 2 ) α   Υ θ ( μ ,   θ ) ] .
For the right side of Equation (111), since Υ ˜ be an U D E T - convex F N V M , we deduce that
Υ ˜ ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 ) F sin π 𝓉 4 e 1 𝓉 Υ ˜ ( μ ) cos π 𝓉 4 e 𝓉 Υ ˜ ( μ + b 2 ) ,
and
Υ ˜ ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ) F cos π 𝓉 4 e 𝓉 Υ ˜ ( μ ) sin π 𝓉 4 e 1 𝓉 Υ ˜ ( μ + b 2 ) .
Adding (118) and (119), we have
Υ ~ 𝓉 μ + 1 𝓉 μ + b 2 Υ ~ 1 𝓉 μ + 𝓉 μ + b 2 F Υ ~ μ Υ ~ μ + b 2 s i n π 𝓉 4 e 1 𝓉 + c o s π 𝓉 4 e 𝓉 .
Since Υ ˜ is F N V M , then, for each   θ [ 0 ,   1 ] , we have
Υ 𝓉 μ + 1 𝓉 μ + b 2 ,   θ + Υ 1 𝓉 μ + 𝓉 μ + b 2 ,   θ Υ μ ,   θ + Υ μ + b 2 ,   θ s i n π 𝓉 4 e 1 𝓉 + c o s π 𝓉 4 e 𝓉 ,   Υ 𝓉 μ + 1 𝓉 μ + b 2 ,   θ + Υ 1 𝓉 μ + 𝓉 μ + b 2 ,   θ Υ μ ,   θ + Υ μ + b 2 ,   θ s i n π 𝓉 4 e 1 𝓉 + c o s π 𝓉 4 e 𝓉 .  
Taking Υ * ( . ,   θ ) from (121) and multiplying the inequality with e ν 𝓉 2 and integrating the results with 𝓉 from 0 to 1 , we have
0 1 e e ν 𝓉 2 Υ * ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 ,   θ   ) d 𝓉 + 0 1 e e ν 𝓉 2 Υ * ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ,   θ ) d 𝓉 [ Υ * ( μ ,   θ ) + Υ * ( μ + b 2 ,   θ ) ] 0 1 e e ν 𝓉 2 [ sin π 𝓉 4 e 1 𝓉 + cos π 𝓉 4 e 𝓉 ] d 𝓉 , = ( 2 ν + 2 π e ν + 2 2 + 4 ν 2 + 4 ν + π 2 + 4 + 2 π e 1 + 2 e ν 2 ( 2 + ν ) ν 2 4 ν + π 2 + 4 ) [ Υ * ( μ ,   θ ) + Υ * ( μ + b 2 ,   θ ) ] .
In a manner identical to that described previously, we have Υ * ( . ,   θ )
0 1 e e ν 𝓉 2 Υ * ( 𝓉 μ + ( 1 𝓉 ) μ + b 2   ,   θ ) d 𝓉 + 0 1 e e ν 𝓉 2 Υ * ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ,   θ ) d 𝓉 [ Υ * ( μ ,   θ ) + Υ * ( b ,   θ ) ] 0 1 e ν 𝓉 [ sin π 𝓉 4 e 2 𝓉 2 + cos π 𝓉 4 e 𝓉 2 ] d 𝓉 , = ( 2 ν + 2 π e ν + 2 2 + 4 ν 2 + 4 ν + π 2 + 4 + 2 π e 1 + 2 e ν 2 ( 2 + ν ) ν 2 4 ν + π 2 + 4 ) [ Υ * ( μ ,   θ ) + Υ * ( μ + b 2 ,   θ ) ] .
From (122) and (123), we have
0 1 e ν 𝓉 Υ θ ( 𝓉 μ + ( 1 𝓉 ) μ + b 2 ) d 𝓉 + 0 1 e ν 𝓉 Υ θ ( ( 1 𝓉 ) μ + 𝓉 μ + b 2 ) d 𝓉 I ( 2 ν + 2 π e ν + 2 2 + 4 ν 2 + 4 ν + π 2 + 4 + 2 π e 1 + 2 e ν 2 ( 2 + ν ) ν 2 4 ν + π 2 + 4 ) [ Υ θ ( μ ) + Υ θ ( μ + b 2 ) ] .
Combining (117) and (124), we have
e 2 Υ θ ( 3 μ + b 2 ) I 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ θ ( μ + b 2 ) + ( μ + b 2 ) α Υ θ ( μ ) ] I ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ θ ( μ ) + Υ θ ( μ + b 2 ) 2 ) ,
where
K ( ν ) = 2 ν + 2 π e ν + 2 2 + 4 ν 2 + 4 ν + π 2 + 4 + 2 π e 1 + 2 e ν 2 ( 2 + ν ) ν 2 4 ν + π 2 + 4 .
Similarly, if we take the interval [ μ + b 2 ,   b ] , then, from (38), we acquire that
e 2 [ Υ θ ( μ + 3 b 2 ) ] I 1 α 2 ( 1 e ν 2 ) [ ( μ + b 2 ) + α   Υ θ ( b ) + b α   Υ θ ( μ + b 2 ) ] I ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ θ ( μ + b 2 ) + Υ θ ( b ) 2 ) .
Adding (125) and (126), we have
e 2 [ Υ θ ( 3 μ + b 2 ) + Υ θ ( μ + 3 b 2 ) ] I 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ θ ( μ + b 2 ) + ( μ + b 2 ) + α   Υ θ ( b ) + ( μ + b 2 ) α Υ θ ( μ ) + b α   Υ θ ( μ + b 2 ) ] I ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ θ ( μ ) + Υ θ ( b ) 2 + Υ θ ( μ + b 2 ) ) .
To achieve the first and fourth fuzzy inclusion relations in (111), we take
Υ θ ( μ + b 2 ) = Υ θ ( 3 μ + b 4 + μ + 3 b 4 2 ) I sin π 4 e Υ θ ( μ ) + cos π 4 e Υ θ ( b ) = 1 2 2 e Υ θ ( μ ) + 1 2 2 e Υ θ ( b ) ,
and
Υ θ ( μ + b 2 ) = Υ θ ( 3 μ + b 4 + μ + 3 b 4 2 ) I sin π 4 e Υ θ ( 3 μ + b 4 ) + cos π 4 e Υ θ ( μ + 3 b 4 ) = 1 2 2 e Υ θ ( 3 μ + b 4 ) + 1 2 2 e Υ θ ( μ + 3 b 4 ) .
By using the inclusion relations in (128) and (129), we obtain the first and fourth fuzzy inclusions of (111). By combining the resultant inclusion and (127), we obtain the following relations
e Υ θ ( μ + b 2 ) I e 2 [ Υ θ ( 3 μ + b 2 ) + Υ θ ( μ + 3 b 2 ) ] I 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ θ ( μ + b 2 ) + ( μ + b 2 ) + α   Υ θ ( b ) + ( μ + b 2 ) α   Υ θ ( μ ) + b α   Υ θ ( μ + b 2 ) ] I ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ θ ( μ ) + Υ θ ( b ) 2 + Υ θ ( μ + b 2 ) ) I ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ θ ( μ ) + Υ θ ( b ) 2 .
That is
e Υ ˜ ( μ + b 2 ) F e 2 [ Υ ˜ ( 3 μ + b 2 ) Υ ˜ ( μ + 3 b 2 ) ] F 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ ˜ ( μ + b 2 ) ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) b α   Υ ˜ ( μ + b 2 ) ] F ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ ˜ ( μ ) Υ ˜ ( b ) 2 Υ ˜ ( μ + b 2 ) ) F ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 ,
hence the required results. □
Remark 8. 
We readily see from Theorem 10 that
If one lays Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then one can acquire (see [78])
e Υ ˜ ( μ + b 2 ) F e 2 [ Υ ˜ ( 3 μ + b 2 ) Υ ˜ ( μ + 3 b 2 ) ] F 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ ˜ ( μ + b 2 ) ( μ + b 2 ) + α   Υ ˜ ( b ) ( μ + b 2 ) α   Υ ˜ ( μ ) b α   Υ ˜ ( μ + b 2 ) ] F ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ ˜ ( μ ) Υ ˜ ( b ) 2 Υ ˜ ( μ + b 2 ) ) F ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
Let Υ * ( ϰ , θ ) Υ * ( ϰ , θ ) and θ = 1 . Then, from Theorem 10, we have (see [86])
e Υ ( μ + b 2 ) e 2 [ Υ ( 3 μ + b 2 ) + Υ ( μ + 3 b 2 ) ] 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ ( μ + b 2 ) + ( μ + b 2 ) + α   Υ ( b ) + ( μ + b 2 ) α   Υ ( μ ) + b α   Υ ( μ + b 2 ) ] ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ ( μ ) + Υ ( b ) 2 + Υ ( μ + b 2 ) ) ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ ( μ ) + Υ ( b ) 2 .
If α 1 , that is
lim α 1 ν = lim α 1 1 α α ( b μ ) = 0 ,
then
lim α 1 ν 1 e ν 2 ( 2 ν + 2 π e ν + 2 2 + 4 ν 2 + 4 ν + π 2 + 4 + 2 π e 1 + 2 e ν 2 ( 2 + ν ) ν 2 4 ν + π 2 + 4 ) = 4 ( 2 π e 1 + 4 ) π 2 + 4 ,   lim α 1 1 α 2 ( 1 e e ν 2 ) = 1 b μ .
The next finding, which is likewise novel, is as follows:
e 2 Υ ˜ ( μ + b 2 ) F 1 2 e 2 [ Υ ˜ ( 3 μ + b 2 ) Υ ˜ ( μ + 3 b 2 ) ] F 1 b μ μ b Υ ˜ ( ϰ ) d ϰ F 2 π e 1 + 4 π 2 + 4 ( Υ ˜ ( μ ) Υ ˜ ( b ) 2 Υ ˜ ( μ + b 2 ) ) F 2 π e 1 + 4 π 2 + 4 ( 1 + e 2 ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
If one lays α 1 and Υ ˜ is a lower U D E T - convex F N V M on [ μ , b ] , then one can achieve (see [78])
e 2 Υ ˜ ( μ + b 2 ) F 1 2 e 2 [ Υ ˜ ( 3 μ + b 2 ) Υ ˜ ( μ + 3 b 2 ) ] F 1 b μ μ b Υ ˜ ( ϰ ) d ϰ F 2 π e 1 + 4 π 2 + 4 ( Υ ˜ ( μ ) Υ ˜ ( b ) 2 Υ ˜ ( μ + b 2 ) ) F 2 π e 1 + 4 π 2 + 4 ( 1 + e 2 ) Υ ˜ ( μ ) Υ ˜ ( b ) 2 .
Let α 1 and Υ * ( ϰ ,   θ ) Υ * ( ϰ ,   θ ) with . Then, from Theorem 10, we have (see [86])
e 2 Υ ( μ + b 2 ) 1 2 e 2 [ Υ ( 3 μ + b 2 ) + Υ ( μ + 3 b 2 ) ] 1 b μ μ b Υ ( ϰ ) d ϰ 2 π e 1 + 4 π 2 + 4 ( Υ ( μ ) + Υ ( b ) 2 + Υ ( μ + b 2 ) ) 2 π e 1 + 4 π 2 + 4 ( 1 + e 2 ) Υ ( μ ) + Υ ( b ) 2 .
If Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) with θ = 1 , then, from Theorem 10, we acquire
e Υ ( μ + b 2 ) e 2 [ Υ ( 3 μ + b 2 ) + Υ ( μ + 3 b 2 ) ] 1 α 2 ( 1 e ν 2 ) [ μ + α   Υ ( μ + b 2 ) + ( μ + b 2 ) + α   Υ ( b ) + ( μ + b 2 ) α   Υ ( μ ) + b α   Υ ( μ + b 2 ) ] ν 2 ( 1 e ν 2 ) K ( ν ) ( Υ ( μ ) + Υ ( b ) 2 + Υ ( μ + b 2 ) ) ν 2 ( 1 e ν 2 ) ( 1 + e 2 ) K ( ν ) Υ ( μ ) + Υ ( b ) 2 .
Let α 1 and Υ * ( ϰ ,   θ ) = Υ * ( ϰ ,   θ ) with θ = 1 . Then, from Theorem 10, we achieve (see [73])
e 2 Υ ( μ + b 2 ) 1 2 e 2 [ Υ ( 3 μ + b 2 ) + Υ ( μ + 3 b 2 ) ] 1 b μ μ b Υ ( ϰ ) d ϰ 2 π e 1 + 4 π 2 + 4 ( Υ ( μ ) + Υ ( b ) 2 + Υ ( μ + b 2 ) ) 2 π e 1 + 4 π 2 + 4 ( 1 + e 2 ) Υ ( μ ) + Υ ( b ) 2 .
Example 3. 
Let  α = 1 3 ,   ϰ [ 0 , 1 ]  and the  F N V M   Υ ˜ : [ μ ,   b ] = [ 0 ,   1 ] R C ,  defined by
Υ ˜ ( ϰ ) ( θ ) = { θ 2 ϰ 4 3 2 2 ϰ 4                 θ [ 2 ϰ 4 , 3 2 ] ,   1 + x θ x 1 2             θ ( 3 2 ,   1 + x ] ,   0                        
then, for each  θ [ 0 ,   1 ] ,  we have  Υ θ ( ϰ ) = [ 2 ( 1 θ ) ϰ 4 + 3 2 θ , ( 1 θ ) ( 1 + x ) + 3 2 θ ] . It can be easily seen that, for each  θ [ 0 ,   1 ] , the left and right end point functions  Υ * ( ϰ , θ ) = 2 ( 1 θ ) ϰ 4 + 3 2 θ ,   Υ * ( ϰ ,   θ ) = ( 1 θ ) ( 1 + x ) + 3 2 θ  are exponential trigonometric convex and concave functions, respectively. Then  Υ ˜ ( ϰ )  is  a   U D E T convex  F N V M . We readily see that  Υ ˜ L ( [ μ ,   b ] , R C )  and  ν = 1 α α ( b μ ) = 2 .
e Υ μ + b 2 , θ = e 1 8 1 θ + 3 2 θ e Υ μ + b 2 , θ = e 3 2 1 θ + 3 2 θ e 2 Υ 3 μ + b 2 , θ + Υ μ + 3 b 2 , θ = e 2 41 64 1 θ + 3 2 θ e 2 Υ 3 μ + b 2 , θ + Υ μ + 3 b 2 , θ = e 2 3 1 θ + 3 2 θ 1 α 2 1 e ν 2 I μ + α Υ μ + b 2 , θ + I μ + b 2 + α Υ b , θ + I μ + b 2 α Υ μ , θ + I b α   Υ μ + b 2 , θ = 1 1 e 1 0 1 2 e 2 1 2 x   2 1 θ ϰ 4 + 3 2 θ d x + 0 1 2 e 2 x 2 1 θ ϰ 4 + 3 2 θ d x + 1 1 e 1 1 2 1 e 2 1 x     2 1 θ ϰ 4 + 3 2 θ d x + 1 2 1 e 2 x 1 2 2 1 θ ϰ 4 + 3 2 θ d x = 1 1 e 1 1 θ 3 e 2 121 e 1 8 + 2 + 12 θ . 1 α 2 1 e ν 2 I μ + α Υ μ + b 2 , θ + I μ + b 2 + α Υ b , θ + I μ + b 2 α Υ μ , θ + I b α   Υ μ + b 2 , θ = 1 1 e 1 0 1 2 e 2 1 2 x     1 θ 1 + x + 3 2 θ d x + 0 1 2 e 2 x 1 θ 1 + x + 3 2 θ d x + 1 1 e 1 1 2 1 e 2 1 x   1 θ 1 + x + 3 2 θ d x + 1 2 1 e 2 x 1 2 1 θ 1 + x + 3 2 θ d x = 1 1 e 1 1 θ 3 3 e 1 + 12 θ ν 2 1 e ν 2 K ν Υ μ , θ + Υ b , θ 2 + Υ μ + b 2 , θ = 1 1 e 1 6 + 2 π e 2 16 + π 2 + 2 e 1 π 9 8 1 θ + 3 θ ν 2 1 e ν 2 K ν Υ μ , θ + Υ b , θ 2 + Υ μ + b 2 , θ = 1 1 e 1 6 + 2 π e 2 16 + π 2 + 2 e 1 π 3 1 θ + 3 θ ν 2 1 e ν 2 1 + e 2 K ν Υ μ , θ + Υ b , θ 2 = 1 1 e 1 1 + e 2 6 + 2 π e 2 16 + π 2 + 2 e 1 π 1 θ + 3 2 θ ν 2 1 e ν 2 1 + e 2 K ν Υ μ , θ + Υ b , θ 2 = 1 1 e 1 1 + e 2 6 + 2 π e 2 16 + π 2 + 2 e 1 π 3 2 1 θ + 3 2 θ e 1 8 1 θ + 3 2 θ , 3 2 1 θ + 3 2 θ e 2 41 64 1 θ + 3 2 θ , 3 1 θ + 3 2 θ I 1 1 e 1 1 θ 3 e 2 121 e 1 8 + 2 + 12 θ , 1 θ 3 3 e 1 + 12 θ I 1 1 e 1 6 + 2 π e 2 16 + π 2 + 2 e 1 π 9 8 1 θ + 3 θ , 3 1 θ + 3 θ I 1 1 e 1 1 + e 2 6 + 2 π e 2 16 + π 2 + 2 e 1 π 1 θ + 3 2 θ , 3 2 1 θ + 3 2 θ ,
Hence, Theorem 10 has been verified.

4. Conclusions

The U D E T convex fuzzy number valued mappings are a novel class of fuzzy convex functions that we introduced in this paper. We also looked at some of their algebraic characteristics. For the newly constructed convex function and fuzzy fractional integral operators, we created a brand new inequality of H H   types. Additionally, for the newly specified convex function, we proved a new midpoint H H   type and a few associated integral inequalities. In Remarks 2–8, we explored a number of particular situations that were discovered as a result of the major findings and made note of their presence in the literature. Finally, we applied our findings to particular values and functions to demonstrate some applications of Bessel functions and special means. In any event, we anticipate that these findings will help us better comprehend the fundamentals of fractional calculus and its various applications.

Author Contributions

Conceptualization, M.B.K.; methodology, M.B.K.; validation, S.A.; formal analysis, S.A.; investigation, M.B.K.; resources, S.A.; data curation, A.A.; writing—original draft preparation, M.B.K.; writing—review and editing, M.B.K. and S.A.; visualization, A.A.; supervision, M.B.K. and J.E.M.-D.; project administration, M.B.K. and J.E.M.-D.; and funding acquisition, J.E.M.-D. and A.A. All authors have read and agreed to the published version of the manuscript.

Funding

The researchers would like to acknowledge Deanship of Scientific Research, Taif University for funding this work.

Data Availability Statement

Not applicable.

Acknowledgments

The authors would like to thank the Rector of COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Budak, H.; Tunç, T.; Sarikaya, M. Fractional Hermite-Hadamard-type inequalities for interval-valued functions. Proc. Am. Math. Soc. 2020, 148, 705–718. [Google Scholar] [CrossRef] [Green Version]
  2. Du, T.-S.; Luo, C.A.; Yu, B. Certain quantum estimates on the parameterized integral inequalities and their applications. J. Math. Inequal. 2021, 15, 201–228. [Google Scholar] [CrossRef]
  3. Du, T.; Wang, H.; Khan, M.A.; Zhang, Y. Certain Integral Inequalities Considering Generalized M-Convexity on Fractal Sets and Their Applications. Fractals 2019, 27, 1950117. [Google Scholar] [CrossRef]
  4. Set, E.; Butt, S.I.; Akdemir, A.O.; Karaoǧlan, A.; Abdeljawad, T. New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators. Chaos Solitons Fractals 2021, 143, 110554. [Google Scholar] [CrossRef]
  5. Budak, H.; Kara, H.; Ali, M.A.; Khan, S.; Chu, Y. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions. Open Math. 2021, 19, 1081–1097. [Google Scholar] [CrossRef]
  6. Zhao, T.; Wang, M.; Chu, Y. On the Bounds of the Perimeter of an Ellipse. Acta Math. Sci. 2022, 42, 491–501. [Google Scholar] [CrossRef]
  7. Zhao, T.-H.; Wang, M.-K.; Hai, G.-J.; Chu, Y.-M. Landen inequalities for Gaussian hypergeometric function. RACSAM Rev. R. Acad. A 2022, 116, 1–23. [Google Scholar] [CrossRef]
  8. Wang, M.-K.; Hong, M.-Y.; Xu, Y.-F.; Shen, Z.-H.; Chu, Y. Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 2020, 14, 1–21. [Google Scholar] [CrossRef]
  9. Zhao, T.-H.; Chu, Y.; Qian, W.-M. Sharp power mean bounds for the tangent and hyperbolic sine means. J. Math. Inequal. 2021, 15, 1459–1472. [Google Scholar] [CrossRef]
  10. Chu, Y.-M.; Wang, G.-D.; Zhang, X.-H. The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 2011, 284, 653–663. [Google Scholar] [CrossRef]
  11. Kórus, P. An extension of the Hermite–Hadamard inequality for convex and s-convex functions. Aequ. Math. 2019, 93, 527–534. [Google Scholar] [CrossRef] [Green Version]
  12. Abramovich, S.; Persson, L.E. Fejér and Hermite–Hadamard type inequalities for N-quasi-convex functions. Math. Notes 2017, 102, 599–609. [Google Scholar] [CrossRef] [Green Version]
  13. Delavar, M.R.; De La Sen, M. A mapping associated to h-convex version of the Hermite–Hadamard inequality with applications. J. Math. Inequal. 2020, 14, 329–335. [Google Scholar] [CrossRef]
  14. Ahmad, B.; Alsaedi, A.; Kirane, M.; Torebek, B.T. Hermite–Hadamard, Hermite–Hadamard–Fejér, Dragomir–Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals. J. Comput. Appl. Math. 2019, 353, 120–129. [Google Scholar] [CrossRef] [Green Version]
  15. Khan, M.A.; Ali, T.; Dragomir, S.S.; Sarikaya, M.Z. Hermite–Hadamard type inequalities for conformable fractional integrals. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 2018, 112, 1033–1048. [Google Scholar] [CrossRef]
  16. Marinescu, D.; Monea, M. A Very Short Proof of the Hermite–Hadamard Inequalities. Am. Math. Mon. 2020, 127, 850–851. [Google Scholar] [CrossRef]
  17. İşcan, İ. Weighted Hermite–Hadamard–Mercer type inequalities for convex functions. Numer. Methods Partial. Differ. Equ. 2021, 37, 118–130. [Google Scholar] [CrossRef]
  18. Kadakal, M.; Karaca, H.; Iscan, I. Hermite-Hadamard Type Inequalities for Multiplicatively Geometrically P-Functions. Poincare J. Anal. Appl. 2018, 5, 77–85. [Google Scholar] [CrossRef] [PubMed]
  19. Kadakal, H.; Bekar, K. New Inequalities for Ah-Convex Functions Using Beta and Hypergeometric Functions. Poincare J. Anal. Appl. 2019, 6, 105–114. [Google Scholar] [CrossRef]
  20. Chu, Y.-M.; Xia, W.-F.; Zhang, X.-H. The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivar. Anal. 2012, 105, 412–421. [Google Scholar] [CrossRef] [Green Version]
  21. Hajiseyedazizi, S.N.; Samei, M.E.; Alzabut, J.; Chu, Y.-M. On multi-step methods for singular fractional q-integro-differential equations. Open Math. 2021, 19, 1378–1405. [Google Scholar] [CrossRef]
  22. Jin, F.; Qian, Z.S.; Chu, Y.M.; Rahman, M. On nonlinear evolution model for drinking behavior under Caputo-Fabrizio deriv-ative. J. Appl. Anal. Comput. 2022, 12, 790–806. [Google Scholar]
  23. Wang, F.; Khan, M.N.; Ahmad, I.; Ahmad, H.; Abu-Zinadah, H.; Chu, Y.-M. Numerical Solution of Traveling Waves in Chemical Kinetics: Time-Fractional Fishers Equations. Fractals 2022, 30, 2240051. [Google Scholar] [CrossRef]
  24. Chu, Y.-M.; Siddiqui, M.K.; Nasir, M. On Topological Co-Indices of Polycyclic Tetrathiafulvalene and Polycyclic Oragano Silicon Dendrimers. Polycycl. Aromat. Compd. 2022, 42, 2179–2197. [Google Scholar] [CrossRef]
  25. Chu, Y.-M.; Rauf, A.; Ishtiaq, M.; Siddiqui, M.K.; Muhammad, M.H. Topological Properties of Polycyclic Aromatic Nanostars Dendrimers. Polycycl. Aromat. Compd. 2022, 42, 1891–1908. [Google Scholar] [CrossRef]
  26. Chu, Y.M.; Numan, M.; Butt, S.I.; Siddiqui, M.K.; Ullah, R.; Cancan, M.; Ali, U. Degree-based topological aspects of polyphe-nylene nanostructures. Polycycl. Aromat. Compd. 2022, 42, 2591–2606. [Google Scholar] [CrossRef]
  27. Chu, Y.-M.; Muhammad, M.H.; Rauf, A.; Ishtiaq, M.; Siddiqui, M.K. Topological Study of Polycyclic Graphite Carbon Nitride. Polycycl. Aromat. Compd. 2022, 42, 3203–3215. [Google Scholar] [CrossRef]
  28. Nwaeze, E.R.; Khan, M.A.; Chu, Y.-M. Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions. Adv. Differ. Equ. 2020, 2020, 1–17. [Google Scholar] [CrossRef]
  29. Zhao, T.-H.; Bhayo, B.A.; Chu, Y.-M. Inequalities for Generalized Grötzsch Ring Function. Comput. Methods Funct. Theory 2022, 22, 559–574. [Google Scholar] [CrossRef]
  30. Zhao, T.-H.; He, Z.-Y.; Chu, Y.-M. Sharp Bounds for the Weighted Hölder Mean of the Zero-Balanced Generalized Complete Elliptic Integrals. Comput. Methods Funct. Theory 2021, 21, 413–426. [Google Scholar] [CrossRef]
  31. Zhao, T.-H.; Wang, M.-K.; Chu, Y. Concavity and bounds involving generalized elliptic integral of the first kind. J. Math. Inequal. 2021, 15, 701–724. [Google Scholar] [CrossRef]
  32. Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. Monotonicity and convexity involving generalized elliptic integral of the first kind. RACSAM Rev. R. Acad. A 2021, 115, 46. [Google Scholar] [CrossRef]
  33. Chu, H.-H.; Zhao, T.-H.; Chu, Y.-M. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means. Math. Slovaca 2020, 70, 1097–1112. [Google Scholar] [CrossRef]
  34. Zhao, T.H.; He, Z.Y.; Chu, Y.M. On some refinements for inequalities involving zero-balanced hyper geometric function. AIMS Math. 2020, 5, 6479–6495. [Google Scholar] [CrossRef]
  35. Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 2020, 5, 4512–4528. [Google Scholar] [CrossRef]
  36. Tariq, M.; Alsalami, O.M.; Shaikh, A.A.; Nonlaopon, K.; Ntouyas, S.K. New Fractional Integral Inequalities Pertaining to Caputo–Fabrizio and Generalized Riemann–Liouville Fractional Integral Operators. Axioms 2022, 11, 618. [Google Scholar] [CrossRef]
  37. Chen, H.; Katugampola, U.N. Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals. J. Math. Anal. Appl. 2017, 446, 1274–1291. [Google Scholar] [CrossRef] [Green Version]
  38. Du, T.; Awan, M.U.; Kashuri, A.; Zhao, S. Some k-fractional extensions of the trapezium inequalities through generalized relative semi-(m, h)-preinvexity. Appl. Anal. 2021, 100, 642–662. [Google Scholar] [CrossRef]
  39. Mehrez, K.; Agarwal, P. New Hermite–Hadamard type integral inequalities for convex functions and their applications. J. Comput. Appl. Math. 2019, 350, 274–285. [Google Scholar] [CrossRef]
  40. Kunt, M.; Iscan, I.; Turhan, S.; Karapinar, D. Improvement of fractional Hermite–Hadamard type inequality for convex functions. Miskolc Math. Notes 2018, 19, 1007–1017. [Google Scholar] [CrossRef]
  41. Kara, H.; Budak, H.; Ali, M.A.; Sarikaya, M.Z.; Chu, Y.-M. Weighted Hermite–Hadamard type inclusions for products of co-ordinated convex interval-valued functions. Adv. Differ. Equ. 2021, 2021, 104. [Google Scholar] [CrossRef]
  42. Chu, Y.; Zhao, T.-H. Concavity of the error function with respect to Hölder means. Math. Inequal. Appl. 2016, 19, 589–595. [Google Scholar] [CrossRef]
  43. Zhao, T.-H.; Shi, L.; Chu, Y.-M. Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. RACSAM Rev. R. Acad. A 2020, 114, 96. [Google Scholar] [CrossRef]
  44. Zhao, T.-H.; Zhou, B.-C.; Wang, M.-K.; Chu, Y.-M. On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, 2019, 42. [Google Scholar] [CrossRef] [Green Version]
  45. Zhao, T.H.; Wang, M.K.; Zhang, W.; Chu, Y.M. Quadratic transformation inequalities for Gaussian hyper geometric function. J. Inequal. Appl. 2018, 2018, 251. [Google Scholar] [CrossRef] [Green Version]
  46. Qian, W.-M.; Chu, H.-H.; Wang, M.-K.; Chu, Y. Sharp inequalities for the Toader mean of order −1 in terms of other bivariate means. J. Math. Inequal. 2022, 16, 127–141. [Google Scholar] [CrossRef]
  47. Zhao, T.-H.; Chu, H.-H.; Chu, Y. Optimal Lehmer mean bounds for the nth power-type Toader means of n = −1, 1, 3. J. Math. Inequal. 2022, 16, 157–168. [Google Scholar] [CrossRef]
  48. Moore, R.E.; Kearfott, R.B.; Michael, J. Cloud, Introduction to Interval Analysis; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA, 2009. [Google Scholar]
  49. Snyder, J.M. Interval analysis for computer graphics. ACM SIGGRAPH Comput. Graph. 1992, 26, 121–130. [Google Scholar] [CrossRef]
  50. Rothwell, E.J.; Cloud, M.J. Automatic Error Analysis Using Intervals. IEEE Trans. Educ. 2012, 55, 9–15. [Google Scholar] [CrossRef]
  51. de Weerdt, E.; Chu, Q.P.; Mulder, J.A. Neural Network Output Optimization Using Interval Analysis. IEEE Trans. Neural Netw. 2009, 20, 638–653. [Google Scholar] [CrossRef] [Green Version]
  52. Ghosh, D.; Debnath, A.K.; Pedrycz, W. A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions. Int. J. Approx. Reason. 2020, 121, 187–205. [Google Scholar] [CrossRef]
  53. Singh, D.; Dar, B.; Kim, D. KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions. Eur. J. Oper. Res. 2016, 254, 29–39. [Google Scholar] [CrossRef]
  54. Wang, M.-K.; Chu, Y.-M. Re_nements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 2017, 37, 607–622. [Google Scholar] [CrossRef]
  55. Wang, M.-K.; Chu, Y. Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 2018, 21, 521–537. [Google Scholar] [CrossRef] [Green Version]
  56. Wang, M.-K.; Chu, H.-H.; Chu, Y.-M. Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals. J. Math. Anal. Appl. 2019, 480, 123388. [Google Scholar] [CrossRef]
  57. Wang, M.-K.; Chu, Y.-M.; Jiang, Y.-P. Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 2016, 46, 679–691. [Google Scholar] [CrossRef] [Green Version]
  58. Wang, M.-K.; Chu, H.-H.; Li, Y.-M.; Chu, Y.-M. Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind. Appl. Anal. Discret. Math. 2020, 14, 255–271. [Google Scholar] [CrossRef]
  59. Wang, M.-K.; Chu, Y.-M.; Qiu, S.-L.; Jiang, Y.-P. Bounds for the perimeter of an ellipse. J. Approx. Theory 2012, 164, 928–937. [Google Scholar] [CrossRef] [Green Version]
  60. Wang, M.-K.; Chu, Y.-M.; Qiu, Y.-F.; Qiu, S.-L. An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 2011, 24, 887–890. [Google Scholar] [CrossRef] [Green Version]
  61. Wang, M.-K.; Chu, Y.; Zhang, E. Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 2019, 22, 601–617. [Google Scholar] [CrossRef] [Green Version]
  62. Wang, M.-K.; Chu, Y.-M.; Zhang, W. Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. 2019, 43, 653–668. [Google Scholar] [CrossRef]
  63. Wang, M.-K.; He, Z.-Y.; Chu, Y.-M. Sharp Power Mean Inequalities for the Generalized Elliptic Integral of the First Kind. Comput. Methods Funct. Theory 2020, 20, 111–124. [Google Scholar] [CrossRef]
  64. Zhao, D.N.; An, T.N.; Ye, G.; Torres, D.F.M. On Hermite-Hadamard type inequalities for harmonical h-convex interval-valued functions. Math. Inequal. Appl. 2019, 33, 1715–1725. [Google Scholar] [CrossRef] [Green Version]
  65. Román-Flores, H.; Chalco-Cano, Y.; Lodwick, W.A. Some integral inequalities for interval-valued functions. Comput. Appl. Math. 2020, 23, 95–105. [Google Scholar] [CrossRef]
  66. Younus, A.; Nisar, O. Convex optimization of interval valued functions on mixed domains. Filomat 2018, 33, 1715–1725. [Google Scholar] [CrossRef] [Green Version]
  67. Khan, M.B.; Noor, M.A.; Noor, K.I.; Chu, Y.M. New Hermite-Hadamard type inequalities for -convex fuzzy-interval-valued functions. Adv. Differ. Equ. 2021, 2021, 6–20. [Google Scholar] [CrossRef]
  68. Sana, G.; Khan, M.B.; Noor, M.A.; Mohammed, P.O.; Chu, Y.-M. Harmonically Convex Fuzzy-Interval-Valued Functions and Fuzzy-Interval Riemann–Liouville Fractional Integral Inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1809–1822. [Google Scholar] [CrossRef]
  69. Khan, M.B.; Noor, M.A.; Shah, N.A.; Abualnaja, K.M.; Botmart, T. Some New Versions of Hermite–Hadamard Integral Ine-qualities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings. Fractal Fract. 2022, 6, 83. [Google Scholar] [CrossRef]
  70. Khan, M.B.; Noor, M.A.; Abdullah, L.; Chu, Y.-M. Some New Classes of Preinvex Fuzzy-Interval-Valued Functions and Inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1403–1418. [Google Scholar] [CrossRef]
  71. Khan, M.B.; Noor, M.A.; Noor, K.I.; Chu, Y.-M. Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1856–1870. [Google Scholar] [CrossRef]
  72. Khan, M.B.; Santos-García, G.; Noor, M.A.; Soliman, M.S. Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities. Chaos Solitons Fractals 2022, 164, 112692. [Google Scholar] [CrossRef]
  73. Kadakal, M.; Işcan, I.; Agarwal, P.; Jleli, M. Exponential trigonometric convex functions and Hermite-Hadamard type inequalities. Math. Slovaca 2021, 71, 43–56. [Google Scholar] [CrossRef]
  74. Allahviranloo, T.; Salahshour, S.; Abbasbandy, S. Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 2012, 16, 297–302. [Google Scholar] [CrossRef]
  75. Diamond, P.; Kloeden, P. Metric Spaces of Fuzzy Sets: Theory and Applications; World Scientific: Singapore, 1994. [Google Scholar] [CrossRef]
  76. Bede, B. Mathematics of Fuzzy Sets and Fuzzy Logic, Volume 295 of Studies in Fuzziness and Soft Computing; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
  77. Khan, M.B.; Cătaş, A.; Aloraini, N.; Soliman, M.S. Some New Versions of Fractional Inequalities for Exponential Trigonometric Convex Mappings via Ordered Relation on Interval-Valued Settings. Fractal Fract. 2023, 7, 223. [Google Scholar] [CrossRef]
  78. Khan, M.B.; Othman, H.A.; Santos-García, G.; Saeed, T.; Soliman, M.S. On fuzzy fractional integral operators having exponential kernels and related certain inequalities for exponential trigonometric convex fuzzy-number valued mappings. Chaos Solitons Fractals 2023, 169, 113274. [Google Scholar] [CrossRef]
  79. Kaleva, O. Fuzzy differential equations. Fuzzy Sets Syst. 1987, 24, 301–317. [Google Scholar] [CrossRef]
  80. Aubin, J.P.; Cellina, A. Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wis-Senschaften; Springer: Berlin/Heidelberg, Germany, 1984. [Google Scholar]
  81. Aubin, J.P.; Frankowska, H. Set-Valued Analysis; Birkhäuser: Boston, MA, USA, 1990. [Google Scholar]
  82. Costa, T.; Román-Flores, H. Some integral inequalities for fuzzy-interval-valued functions. Inf. Sci. 2017, 420, 110–125. [Google Scholar] [CrossRef]
  83. Costa, T.M. Jensen’s inequality type integral for fuzzy-interval-valued functions. Fuzzy Sets Syst. 2017, 327, 31–47. [Google Scholar] [CrossRef]
  84. Zhang, D.; Guo, C.; Chen, D.; Wang, G. Jensen’s inequalities for set-valued and fuzzy set-valued functions. Fuzzy Sets Syst. 2020, 404, 178–204. [Google Scholar] [CrossRef]
  85. Nanda, N.; Kar, K. Convex fuzzy mappings. Fuzzy Sets Syst. 1992, 48, 129–132. [Google Scholar] [CrossRef]
  86. Zhou, T.U.; Du, T.-S. Certain fractional integral inclusions pertaining to interval-valued exponential trigonometric convex functions. J. Math. Inequal. 2022, 17, 283–314. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Khan, M.B.; Macías-Díaz, J.E.; Althobaiti, A.; Althobaiti, S. Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels. Fractal Fract. 2023, 7, 567. https://doi.org/10.3390/fractalfract7070567

AMA Style

Khan MB, Macías-Díaz JE, Althobaiti A, Althobaiti S. Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels. Fractal and Fractional. 2023; 7(7):567. https://doi.org/10.3390/fractalfract7070567

Chicago/Turabian Style

Khan, Muhammad Bilal, Jorge E. Macías-Díaz, Ali Althobaiti, and Saad Althobaiti. 2023. "Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels" Fractal and Fractional 7, no. 7: 567. https://doi.org/10.3390/fractalfract7070567

APA Style

Khan, M. B., Macías-Díaz, J. E., Althobaiti, A., & Althobaiti, S. (2023). Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels. Fractal and Fractional, 7(7), 567. https://doi.org/10.3390/fractalfract7070567

Article Metrics

Back to TopTop