New Hermite–Hadamard and Jensen Inequalities for Log-H-Convex

International Journal of Computational Intelligence Systems (2021) 14:155 https://doi.org/10.1007/s44196-021-00004-1

RESEARCH ARTICLE

New Hermite–Hadamard and Jensen Inequalities for Log‑h‑Convex Fuzzy‑Interval‑Valued Functions

Muhammad Bilal Khan1 · Lazim Abdullah2 · Muhammad Aslam Noor1 · Khalida Inayat Noor1

Received: 9 December 2020 / Accepted: 23 July 2021 © The Author(s) 2021

Abstract In the preset study, we introduce the new class of convex fuzzy-interval-valued functions which is called log-h-convex fuzzy-interval-valued functions (log-h-convex FIVFs) by means of fuzzy order relation. We have also investigated some properties of log-h-convex FIVFs. Using this class, we present Jensen and Hermite–Hadamard inequalities (HH-inequalities). Moreover, some useful examples are presented to verify HH-inequalities for log-h-convex FIVFs. Several new and known special results are also discussed which can be viewed as an application of this concept.

Keywords Fuzzy-interval-valued functions · Log-h-convex · Hermite–Hadamard inequality · Hemite–Hadamard–Fejér inequality · Jensen’s inequality

1 Introduction was independently introduced by Hadamard, see [24, 25, 31]. It can be expressed in the following way: The theory of convexity in pure and applied sciences has Let Ψ∶K → ℝ be a convex function on a convex set K become a rich source of inspiration. In several branches of and , ∈ K with ≤ . Then, mathematical and engineering sciences, this theory had not only inspired new and profound results, but also ofers a + 1 Ψ ≤ Ψ()d coherent and general basis for studying a wide range of prob- 2 − � lems. Many new notions of convexity have been developed Ψ( )+Ψ( ) and investigated for convex functions and convex sets. Vari- ≤ . ous integral inequalities for convex functions and their vari- 2 (1) ant forms are being constructed using unique and imagina- Fejér [21] introduced HH-Fejér inequality which is major tive concepts and methodologies. Every function is convex generalizations of HH-inequality. It can be expressed as if and only if it fulflls the HH-inequality, which is a type of follows: integral inequality. Hermite presented this inequality, which Let Ψ∶K → ℝ be a convex function on a convex set K and , ∈ K with ≤ and ∇∶[ , ] → ℝ, ∇() ≥ 0, + * , Lazim Abdullah symmetric with respect to 2 then [email protected] + 1 Muhammad Bilal Khan Ψ ≤ Ψ()∇()d [email protected] 2 ∫ ∇( )d � Muhammad Aslam Noor [email protected] Ψ( )+Ψ( ) ≤ ∇()d, Khalida Inayat Noor 2 � [email protected] (2)

1 Department of Mathematics, COMSATS University If ∇(x) = 1 , then (1) is obtained from (2). Similarly, sev- Islamabad, Islamabad, Pakistan eral inequalities can be obtained from (2) by taking diferent 2 Management Science Research Group, Faculty of Ocean values of symmetric function ∇(x). Engineering Technology and Informatics, University Malaysia Terengganu, Terengganu, Malaysia

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It is commonly known that log-convex functions play artifcial intelligence, control engineering, and decision sci- a signifcant role in convex theory since they allow us to ences [26, 28, 30]. derive more precise inequalities than convex functions. The concepts of convexity and generalized convexity are Noor et al. [35] presented following HH-inequality for very crucial in fuzzy optimization. The defnition of fuzzy log-h-convex functions. mapping was frstly introduced by Chang and Zadeh [10]. Let Ψ∶K → ℝ be a convex function on a convex set K Since then, fuzzy mapping has been widely researched by and , ∈ K with ≤ . Then, several scholars. In 1992, Nanda and Kar [33] suggested an idea of convex fuzzy mapping, showing that a fuzzy map- 1 1 + 2h( ) 1 ping is convex if and only if a convex set is its epigraph. By Ψ 2 ≤ exp ln Ψ()d 2 − � considering the defnition of ordering suggested by Goets- � � ⎡ ⎤ chael-Voxman [22], Yan-Xu [47] addressed the convexity and ⎢ 1 ⎥ ⎢ ∫ h()d ⎥ quasicovexity of fuzzy mappings. The class of fuzzy preinvex ≤ [Ψ( ⎣)Ψ( )]0 . ⎦ (3) functions and fuzzy log-preinvex was presented by Noor [34], Ψ and some properties of fuzzy preinvex fuzzy functions were If is concave, then (3) is reversed. obtained. For fuzzy mapping of one variable, Syau [41] intro- Some writers recently studied the diferent classes of log- duced the concepts of pseudoconvexity, invexity and pseu- convex and generalized log-convex functions see [16–20, 36, doinvexity by using the concept of diferentiability and the 38, 43, 46] and the references therein. results provided by Goetschel and Voxman [22]. A new notion One of these inequalities for convex functions is the of noncovex fuzzy mapping, which is known as B-vex fuzzy Jensen inequality [1, 27], which may be written as follows. mapping, was proposed and explored by Syau [42]. The appli- ∈ [0, 1] u ∈ [ , ], (j = 1, 2, 3, … k, k ≥ 2) Ψ Let j , j and cation to convex fuzzy programing was considered by Wang be a convex function, then and Wu [44] by defning the fuzzy subdiferential of a fuzzy k k mapping. Wu-Xu [45] presented the notions of fuzzy pseudo- Ψ jzj ≤ jΨ zj , (4) convex, fuzzy pseudoinvex, fuzzy invex and fuzzy preinvex j=1 j=1 mapping from “n” dimensional Euclidean space to the set of k fuzzy numbers depending upon the Wang-Wu [44] defnitions with = j = 1. If Ψ is concave, then (4) is reversed. j 1 of diferentiability of fuzzy mapping. Moreover, several prop- Moore [32] explored the fundamental principles of inter- ∑ erties were investigated. Refer to [3, 7, 22, 26, 28, 30] for more val analysis, and Kulish and Miranker [29] looked into the studies on convexity and nonconvexity for fuzzy mappings. fundamental properties and defned the partial order rela- There are some integrals that deal with FIVFs, with tion between intervals. Recently, Guo et al. [23] proposed FIVFs as the integrands. For example, Oseuna-Gomez et al. the defnition of log-h-convex interval-valued functions (in [37] and Costa et al. [12] built Jensen’s integral inequality short, log-h-convex-IVF) and proved the following HH- for FIVFs. Costa and Floures used the same approach to inequality for log-h-convex IVFs: ℝ → + show Minkowski and Beckenbach’s inequalities, where the Let Ψ∶[ , �] ⊂ C be a log-h-convex-IVF given ∗ integrands are FIVFs. Inspired by [11, 12, 23, 37], we gen- by Ψ( ) = Ψ∗( ), Ψ ( ) for all ∈ [ , ] . If Ψ is Riemann eralize integral inequality (1), (2), and (3) by constructing integrable, then fuzzy-interval integral inequality for convex fuzzy-interval- � 1 valued functions (convex FIVF), where the integrands are + � 2h 1 1 Ψ ( 2 ) ⊇ Ψ() convex FIVFs, utilizing this notion on fuzzy-interval space. exp ln d 2 � − � Motivated and inspired by the above literature, we � � ⎡ ⎤ ⎢ 1 ⎥ consider new class of convex FIVFs, which is called log- ⎢ ∫ h(�)d� ⎥ h-convex FIVFs. By using this class, we discuss integral ine- ⊇ [Ψ( ⎣)Ψ(�)]0 . ⎦ (5) qualities (2) and (3) by constructing fuzzy-interval integral We encourage readers to go more into the literature on inequalities, which are known as fuzzy-interval HH-integral generalized convex functions and HH-inequality, particu- inequality and HH-Fejér integral inequality. For log-h-con- larly [2, 4–6, 8, 9, 13, 20, 40, 49–55] and the references vex FIVFs, some Jensen inequalities are also introduced. therein. The theory of fuzzy sets and systems has progressed in a variety of ways from its introduction fve decades ago, as 2 Preliminaries seen in [48]. As a result, it is useful in the study of a variety of problems in pure mathematics and applied sciences, such as In this section, we recall some basic preliminary notions, operation research, computer science, management sciences, defnitions and results. With the help of these results, some new basic defnitions and results are also discussed.

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We begin by recalling the basic notations and defnitions. acceptable in this study. � (ℝ) also stand for the collection of We defne interval as, all fuzzy subsets of ℝ. A real fuzzy interval is a fuzzy set in ℝ with the follow- , ∗ = ∈ ℝ ∶ ≤ ≤ ∗ and , ∗ ∈ ℝ , ∗ ∗ ∗ ing properties: ∗ where ∗ ≤ . ℝ ∗ ∗ ∗ (1) is normal, i.e., there exists ∈ such that ( ) = 1; We write len ∗, = − ∗ . If len ∗, = 0 , ℝ ∗ (2) is upper semi continuous, i.e., for given ∈ , for then ∗, is called degenerate. In this article, all inter- every ∈ ℝ there exist �>0 there exist �>0 such vals will be non-degenerate intervals. The collection ℝ ℝ that �( ) − �(y) <� for all y ∈ with − y <�. of all closed and bounded intervals of is denoted and ∗ ∗ ℝ ∗ (3) is fuzzy convex, i.e., ((1 − ) + y) ≥ min( ( ), (y)), ∀ defined as KC = [ ∗, ]∶ ∗, ∈ and ∗ ≤ . ℝ ∗ , y ∈ and ∈ [0, 1]; If ∗ ≥ 0 , then ∗, is called positive interval. The ℝ + (4) is compactly supported, i.e., cl{ ∈ ( ) 0} is set of all positive interval is denoted by KC and defned as + ∗ ∗ compact. K = ∗, ∶ ∗, ∈ KC and ∗ ≥ 0 . C � ⟩ We’ll now look at some of the properties of intervals ∗ ∗ The collection of all real fuzzy intervals is denoted by 0. using arithmetic operations. Let ∗, , ∗, ∈ C and ℝ Let ∈ 0 be real fuzzy interval, if and only if, -levels ∈ , then we have [ ] is a nonempty compact convex set of ℝ . This is repre- ∗ ∗ ∗ ∗ ∗, + ∗, = ∗ + ∗, + , sented by [ ] = { ∈ ℝ ( ) ≥ }, min , ∗ , ∗, ∗ ∗ , , ∗ × , ∗ = ∗ ∗ ∗ ∗ ∗ ∗ max , ∗ , ∗, ∗ ∗ from these defnitions, we have ∗ ∗ ∗ ∗ ∗ [ ] = ∗(), () , �� , ��∗ if � ≥ 0, �. � , �∗ = ∗ ∗ ��∗, �� if �<0. where ∗ () = inf { ∈ ℝ ( ) ≥ }, , ∗ , , ∗ ∈ K , ⊆ ∗ For ∗ ∗ C the inclusion “ ” is defned by ∗ ∗ ∗ ∗ ∗ () = sup { ∈ ℝ ( ) ≥ }. �∗, � ⊆ ∗, if and only if ∗ ≤ ∗ , ≤ . Thus, a real fuzzy interval can be identifed by a para- Remark 2.1 The relation “ ≤I ” defined on C by ∗ ∗ ∗ ∗ metrized triples ∗, ≤I ∗, if and only if ∗ ≤ ∗, ≤ , for all ∗ ∗ ∗ ∗, , ∗, ∈ C, it is an order relation, see [29]. For ∗(), (), ∶ ∈ [0, 1] . ∗ ∗ ∗ ∗ given ∗, , ∗, ∈ C, we say that ∗, ≤I ∗, ∗ ∗ ∗ ∗ ∗ ≤ , ≤ � ≤ , � < These two end point functions ∗() and () are used to if and only if ∗ ∗ or ∗ ∗ . characterize a real fuzzy interval as a result. Moore [32] initially proposed the concept of Riemann Θ∈ integral for IVF, which is defned as follows: Proposition 2.3 [11] Let , 0 . Then, fuzzy order relation “≼ ” given on 0 by �≼Θ if and only if [ ] ≤I [Θ] for all ℝ → ∈ (0, 1], Theorem 2.2 [32] If Ψ∶[ , �] ⊂ C is an IVF on it is partial order relation. ∗ such that Ψ( ) = Ψ∗, Ψ . Then, Ψ is Riemann integrable ∗ over [ , ] if and only if Ψ∗ and Ψ both are Riemann inte- We’ll now look at some of the properties of fuzzy inter- Θ∈ ∈ ℝ grable over [ , ] such that vals using arithmetic operations. Let , 0 and , then we have (IR) Ψ()d �+Θ̃ = [�] + [Θ] , (7) [�×Θ̃ ] = [�] × [Θ] , (8) = (R) Ψ ()d, (R) Ψ∗()d . ∗ ⎡ ⎤ (6) ⎢ ⎥ [ .] = .[] (9) ⎢ ℝ ⎥ Let be the set of real numbers. A mapping ̃ ⎣ ⎦ For ∈ 0 such that � =Θ+�, we have the existence ∶ ℝ → [0, 1] called the membership function distinguishes of the Hukuhara diference of and Θ , which we call the a fuzzy subset set of ℝ . This representation is found to be

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H-diference of and Θ and denoted by �̃−Θ . If H-diference exists, then (FR) Ψ()d (�)∗(�) = (�̃−Θ)∗(�) = � ∗(�) −Θ∗(�), ⎡ ⎤ (10) ⎢ ⎥ (�)∗(�) = (�̃−Θ)∗(�) = �∗(�) −Θ∗(�). ⎢ ⎥ ⎣ = (R) Ψ (⎦, )d, (R) Ψ∗(, )d ∗ ⎡ ⎤ Theorem 2.4 [14, 39] The space 0 dealing with a supre- ⎢ ⎥ , Θ∈ ⎢ ⎥ mum metric, i.e., for 0. = ⎣(IR) Ψ ()d, ⎦ D( , Θ) = sup H [] , [Θ] , (12) 0≤≤1 ℱℛ for all ∈ (0, 1]. For all ∈ (0, 1], ([�, ],) denotes the it is a complete metric space, where H denotes the well- collection of all FR-integrable FIVFs over [ , ]. known Hausdorf metric on space of intervals. ℝ → � Theorem 2.8 Let Ψ∶[ , �] ⊂ 0 be a FIVF, whose ℝ → Definition 2.5 [11] A fuzzy-interval-valued map -levels defne the family of IVFs Ψ� ∶ [ , �] ⊂ C are Ψ∶K ⊂ ℝ → � ∈ (0, 1], ∗ 0 is called FIVF. For each whose given by Ψ ( ) = Ψ∗( , ), Ψ ( , ) for all ∈ [ , ] and ℝ → -levels define the family of IVFs Ψ� ∶ K ⊂ C for all ∈ (0, 1]. Then, Ψ is (FR)-integrable over [ , ] if Ψ ( )= Ψ ( , ), Ψ∗( , ) ∈ K. ∗ are given by ∗ for all and only if Ψ∗( , ) and Ψ ( , ) both are R-integrable over Here, for each ∈ (0, 1], the end point real functions [ , ] . Moreover, if Ψ is (FR)-integrable over [ , ], then ∗ → ℝ Ψ∗(., ), Ψ (., ) ∶ K are called lower and upper functions of Ψ. (FR) Ψ()d The following conclusions can be drawn from the preced- ⎡ ⎤ ing literature review [11, 14, 28, 32]: ⎢ ⎥ ⎢ ⎥ ∗ ⎣ = (R) Ψ∗( ⎦, )d , (R) Ψ ( , )d Defnition 2.6 Ψ∶[ , �] ⊂ ℝ → � Let 0 be a FIVF. Then, ⎡ ⎤ fuzzy integral of Ψ over [ , ] denoted by (FR) ∫ Ψ()d , ⎢ ⎥ ⎢ ⎥ = ⎣(IR) Ψ ()d, ⎦ and it is given level-wise by (13) (FR) Ψ(�)d� = (IR) Ψ (�)d� for all ∈ (0, 1]. ⎡ � ⎤ � ⎢ ⎥ Defnition 2.9 [38] A function Ψ∶K → ℝ is said to be log- ⎢ ⎥ ⎣ ⎦ � � � ℛ convex function if = Ψ( , )d ∶Ψ( , ) ∈ ([�, ],) , 1− ⎧� ⎫ Ψ( + (1 − )y) ≤ Ψ( ) ×Ψ(y) , ⎪ ⎪ (14) ⎨ ⎬ ⎪ (11)⎪ ∀ , y ∈ K, ∈ [0, 1], where Ψ( ) ≥ 0. If (14) is reversed, ⎩ ℛ ⎭ for all ∈ (0, 1], where ([�, ],) denotes the collection of then Ψ is called log-concave. Ψ Riemannian integrable functions of IVFs. is FR-integrable Ψ∶K → ℝ [ , ] (FR) Ψ( )d ∈ 0. Defnition 2.10 [35] A function is said to be over if Note that, if both end log-h-convex function if point functions are Lebesgue-integrable, then Ψ is fuzzy [ , ] h( ) h(1− ) Aumann-integrable function over , see [28, 32]. Ψ( + (1 − )y) ≤ Ψ( ) ×Ψ(y) , (15) ℝ → � ∀ , y ∈ K, ∈ [0, 1], Ψ( ) 0. Theorem 2.7 Let Ψ∶[ , �] ⊂ 0 be a FIVF, whose where ≥ If (15) is reversed, ℝ → Ψ h -levels defne the family of IVFs Ψ� ∶ [ , �] ⊂ C then is called log- -concave. ∗ h ∶ L → ℝ are given by Ψ ( ) = Ψ∗( , ), Ψ ( , ) for all ∈ [ , ] A function is called super multiplicative if for and for all ∈ (0, 1]. Then, Ψ is FR-integrable over [ , ] all z, y ∈ L , we have ∗ if and only if Ψ∗( , ) and Ψ ( , ) both are R-integrable h(zy) ≥ h( )h(y). (16) over [ , ] . Moreover, if Ψ is FR-integrable over [ , ], then

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h ∗ If (16) is reversed, then is known as sub multiplicative. Ψ∗( + (1 − )y, ), Ψ ( + (1 − )y, ) If the equality hold in (16), then h is called multiplicative. h( ) ∗ h( ) ≤I Ψ∗( , ) , Ψ ( , ) (23) h(1− ) ∗ h(1− ) Definition 2.11 Let K be a convex set and × Ψ ∗(y, ) , Ψ (y, ) . h ∶ [0, 1] ⊆ K → ℝ+ h 0 such that ≢ . Then, FIVF → It follows that Ψ∶K 0 is said to be: h( ) h(1− ) Ψ∗( + (1 − )y, ) ≤ Ψ∗( , ) Ψ∗(y, ) ,

• log-h-convex on K if and Ψ(� + (1 − �)y)≼Ψ( )h(�)×Ψ̃ (y)h(1−�), (17) Ψ∗( + (1 − )y, ) ≤ Ψ∗( , )h( )Ψ∗(y, )h(1− ), ̃ for all , y ∈ K, ∈[0, 1], where Ψ( )≽0. hence, the result has been proved.

• log-h-concave on K if inequality (17) is reversed. ∗ Conversely, suppose that Ψ∗( , ) and Ψ ( , ) both are • Afne log-h-convex on K if log-h-convex functions. Then, from definition and above h(�) h(1−�) Ψ( ) h Ψ(� + (1 − �)y) =Ψ( ) ×Ψ̃ (y) , (18) (23), it follows that is log- -convex FIVF. ̃ for all , y ∈ K, ∈ [0, 1], where Ψ( )≽0. Example 2.14 We consider h( ) ≡ ( ≥ 1) , for ∈ [0, 1] s → Remark 2.12 If h( ) = , then (17) reduces to: and the FIVF Ψ∶[1, 4] 0 is given by,

s s � (1−�) 2 Ψ(� + (1 − �)y)≼Ψ( ) ×Ψ̃ (y) . (19) ∈ 0, ez ez2 2 Ψ( )( ) = 2ez − z2 z2 ( ) = 2 ∈ e ,2e If h , then (17) reduces to: ez � � ⎧ 0 otherwise, Ψ(� + (1 − �)y)≼Ψ( )�×Ψ̃ (y)1−�. ⎪ � � (20) ⎨ Then, ⎪ for each ∈ (0, 1], we have h( ) 1, 2 2 If ≡ then (17) reduces to: ⎩z z ∗ Ψ ( ) = e , (2 − )e . Since Ψ∗( , ) and Ψ ( , ) are Ψ(� + (1 − �)y)≼Ψ( )×Ψ̃ (y). h ∈ (0, 1] (21) log- -convex functions for each then, by Theo- rem 2.13, Ψ( ) is log-h-convex FIVF. Note that, Remarks (i) and (iii) are also new ones. Theorem 2.15 Let K be convex set, non-negative real Ψ ∶ K ⊂ ℝ → + ⊂ And � C C represent the family of valued function h ∶ [0, 1] ⊆ K → ℝ such that h ≢ 0 . Let IVFs through -levels are defne by → ℝ → + Ψ∶K 0 be a FIVF, and Ψ� ∶ K ⊂ C ⊂ C represent the family of IVFs through -levels are given by Theorem 2.13 Let K be convex set and non-nega- ∗ Ψ ( ) = Ψ∗( , ), Ψ ( , ) , ∶ [ ] ⊆ → ℝ tive real valued function h 0, 1 K such that for all ∈ K and for all ∈ (0, 1] Then, Ψ is log-h con- → ̃ . - h ≢ 0, Let Ψ∶K 0 be a FIVF with Ψ( )≽0 , and K, ∈ [0, 1], Ψ ( , ) + cave on if and only if, for all ∗ and Ψ ∶ K ⊂ ℝ → ⊂ represent the family of IVFs ∗ � C C Ψ ( , ) are log-h-concave. through -levels are given by ∗ Proof The proof is similar to that of Theorem 6. Ψ ( ) = Ψ∗( , ), Ψ ( , ) , (22) h( ) = , ∈ [0, 1] for all ∈ K and for all ∈ (0, 1] . Then, Ψ is log-h-convex Example 2.16 We consider for and the ∗ Ψ∶[ , ] = [0, 1] → on K, if and only if, for all ∈ (0, 1], Ψ∗( , ) and Ψ ( , ) FIVFs 0 is given by, are log-h-convex. , ∈ [0, ], Ψ( )( ) = 2 − Proof Let Ψ is log-h-convex FIVF on K. Then, for all , ∈ ( ,2 ], , y ∈ K ∈ (0, 1], ⎧ 0, otherwise, and we have ⎪ Ψ(� + ( − �) )≼Ψ( )h(�)×Ψ̃ ( )h(1−�) ⎨ 1 y y . ⎪ ∈ [0, 1], Then, for⎩ each we have Therefore, from (22) and Proposition 2.3, we have Ψ ( ) = [ , (2 − ) ].

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∗ Since Ψ∗( , ) = z , and Ψ ( , ) = (2 − )z log-h-con- 1 + ln Ψ∗ , ≤ ln Ψ∗( + (1 − ) , ) ∈ [0, 1] 1 vex functions, for each then, by Theorem 2.15, h 2 Ψ( ) is log-h-concave FIVF. 2 + ln Ψ∗((1 − ) + , ), 1 + ln Ψ∗ , ≤ ln Ψ∗( + (1 − ) , ) h 1 2 3 Main Results 2 + ln Ψ∗((1 − ) + , ), First of all, we prove that the following Hermite–Had- amard-type inequality for log-h-convex FIVF. Then, 1 1 + ln , d Ψ∶[ ] → Ψ∗ Theorem 3.1 Let , 0 be a log-h-convex FIVF h 1 � 2 with non-negative real valued function h ∶ [0, 1] → ℝ and 2 0 h 1 0 ∈ [0, 1] Ψ ∶ K ⊂ ℝ → + ⊂ 1 2 ≢ , and for all , � C C ≤ ln Ψ ( + (1 − ) , )d represent the family of IVFs through -levels. If � ∗ ℱℛ Ψ∈ ([�, ],), then 0 1 � 1 1 + � 2h( ) 1 + ln Ψ∗((1 − ) + , )d Ψ 2 ≼ exp (FR) ln Ψ()d � 2 � − � 0 � � ⎡ ⎤ 1 ⎢ 1 ⎥ 1 + ⎢ ∫ h(�)d� ⎥ ln Ψ∗ , d ≼[Ψ( ⎣)×Ψ̃ (�)]0 . ⎦ (24) h 1 � 2 2 0 Ψ h If is log- -concave FIVF then 1 � ∗ 1 ≤ ln Ψ ( + (1 − ) , )d 1 + � 2h( ) 1 � Ψ 2 ≽ exp (FR) ln Ψ()d 0 2 � − � � � ⎡ ⎤ 1 ⎢ 1 ⎥ + ln Ψ∗((1 − ) + , )d. ⎢ ∫ h(�)d� ⎥ 0 � ≽[Ψ( ⎣)×Ψ̃ (�)] . ⎦ 0 → It follows that Proof Let Ψ∶[ , ] 0, log-h-convex FIVF. Then, by Theorem 2.13 and by hypothesis, we have 1 + 1 ln Ψ∗ , ≤ ln Ψ∗( , )d , + � h 1 h 1 1 2 − � Ψ ≼[Ψ(� + (1 − �)�)] 2 ×̃ [Ψ((1 − �) + ��)] 2 . 2h 2 2 1 + 1 Therefore, for every ∈ [0, 1] , we have Ψ∗ Ψ∗( ) ln , ≤ ln , d , 2h 1 2 − � + h 1 2 2 Ψ∗ , ≤ Ψ∗( + (1 − ) , ) 2 h 1 which implies that 2 × Ψ∗((1 − ) + , ) 1 1 + h 1 ∗ ∗ 2 + 2h( ) 1 Ψ , ≤ Ψ ( + (1 − ) , ) Ψ , 2 ≤ exp ln Ψ (, )d , 2 ∗ 2 − � ∗ 1 ∗ h � � ⎛ ⎞ × Ψ ((1 − ) + , ) 2 . ⎜ ⎟ 1 (25) 1 ∗ + 2h( ) ⎜ 1 ∗ ⎟ Ψ , 2 ≤ exp ⎝ ln Ψ (, )d⎠, From (25), we have 2 − � � � ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ That is ⎝ ⎠

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1 1 ∗ 1 1 Ψ∗( , ) =Ψ (, ) = 1 + 2h ∗ + 2h If with , then (24) reduces to, Ψ , ( 2 ) , Ψ , ( 2 ) ∗ 2 2 see [35]: � � � � � � � 1 1 1 exp ∫ ln Ψ∗( , )d , + � 2h( ) 1 − Ψ 2 ≤ exp (R) ln Ψ()d ≤I � � . 2 � − � ⎡ 1 ∗ ⎤ � � ⎡ ⎤ ⎢ exp ∫ ln Ψ ( , )d ⎥ − ⎢ 1 ⎥ ⎢ � � ⎥ ⎢ ∫ h(�)d� ⎥ ⎢ ⎥ ≤ [Ψ( ⎣)×Ψ̃ (�)]0 . ⎦ Thus,⎢ ⎥ ⎣ ⎦ ∗ s If Ψ∗( , ) =Ψ (, ) with = 1 and h( ) = , then (24) � 1 reduces to, see [35]: + � 2h 1 1 Ψ ( 2 ) ≼ exp (FR) ln Ψ()d . 2 � − (26) � � � ⎡ ⎤ + � 2s−1 1 Ψ ≤ exp (R) ln Ψ()d ⎢ ⎥ � − ⎢ ⎥ 2 � In a similar way as above,⎣ we have ⎦ � � ⎡ ⎤ ⎢ 1 ⎥ � ≤ [Ψ( ⎢ )×Ψ̃ (�)] s+1 . ⎥ 1 ⎣ ⎦ exp (FR) ln Ψ( )d ∗ � − � If Ψ∗( , ) =Ψ (, ) with = 1 and h( ) = , then (24) ⎡ ⎤ reduces to, see [15]: ⎢ 1 ⎥ ⎢ ∫ h(�)d� ⎥ (27) � ≼[Ψ( )×Ψ̃ (�)]0 . ⎣ ⎦ + � 1 Ψ ≤ exp (R) ln Ψ()d 2 � − � Combining (26) and (27), we have � � ⎡ ⎤ � ⎢ ̃ ⎥ 1 ≤ Ψ(⎢ )×Ψ(�). ⎥ + � 2h 1 1 Ψ ( 2 ) ≼ exp (FR) ln Ψ()d ⎣ ⎦ 2 � − √ ∗ � If Ψ∗( , ) =Ψ (, ) with = 1 and h( ) ≡ 1 then (24) � � ⎡ ⎤ ⎢ 1 ⎥ reduces to, see [35]: ⎢ ∫ h(�)d� ⎥ 0 � ≼[Ψ( ⎣)×Ψ̃ (�)] . ⎦ 1 + � 2 1 Ψ ≤ exp (R) ln Ψ()d the required result. 2 � − � � � ⎡ ⎤ s Ψ( ⎢)×Ψ̃ (�). ⎥ Remark 3.2 If h( ) = , then (24) reduces to: ≤ ⎢ ⎥ ⎣ ⎦ � Note that, Remarks (i)–(iii) are also new ones. + � 2s−1 1 Ψ ≼ exp (FR) ln Ψ()d 2 � − Example 3.3 h( ) = , ∈ [0, 1] ⎡ ⎤ We consider for , � � Ψ∶[ ] = [ ] → ⎢ 1 ⎥ and the FIVF , 1, 4 0 is given by, ≼[Ψ( )×Ψ̃ (�)] s+1 . z2 z2 ⎢ ⎥ Ψ ( ) = e , (2 − )e , as in Example 2.14, then ⎣ ⎦ z2 Ψ( ) is log-h-convex FIVF. Since, Ψ∗( , ) = e and If h( ) = , then (24) reduces to: ∗ z2 Ψ ( , ) = (2 − )e then, we have � 1 1 2 1 + � 1 5 2h 25 + 2h 1 ( 2 ) Ψ ≼ exp (FR) ln Ψ()d ( 2 ) 2 Ψ∗ , = e = e 4 , 2 � − 2 � � � � � � ⎡ ⎤ � � 4 ⎢ ⎥ 1 1 2 ≼ Ψ(⎢ )×Ψ̃ (�). ⎥ exp ln Ψ (, )d = exp ln ez d = eln ()+7, ⎣ ⎦ − ∗ 3 √ ⎛ ⎞ ⎛ 1 � � ⎞ If h( ) ≡ 1 , then (24) reduces to: ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ � ⎝ 1 ⎠ ⎝ ⎠ 1 () + � 2 1 ∫ h d Ψ ≼ exp (FR) ln Ψ()d Ψ ( , ) ×Ψ (, ) 0 2 � − ∗ ∗ 1 17 � � ⎡ ⎤ 16 = ( e) 4 e 2 = 2 e 2 , ≼Ψ( ⎢)×Ψ̃ (�). ⎥ ⎢ ⎥ ⎣ ⎦ for all ∈ [0, 1]. That means

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25 ln ( )+7 17 Theorem 3.4 Let Ψ∶[ , ] → be a log-h-con- 4 2 0 e ≤ e ≤ 2 e . + vex FIVF with u <� and h ∶ [0, 1] → ℝ , for all ℝ → + Now, we fnd the following inequality ∈ [0, 1] , Ψ� ∶ K ⊂ C ⊂ C represent the fam- ℱℛ ily of IVFs through -levels. If Ψ∈ ([�, ],) and + 1 → ℝ 1 ∇∶[ , ] , ∇( ) ≥ 0, symmetric with respect to , ∗ + 2h( ) 1 ∗ 2 Ψ , 2 ≤ exp ln Ψ (, )d 2 − � then � � ⎡ ⎤ � ⎢ 1 ⎥ ∫ h()d 1 ∗ ⎢ ∗ ⎥ (FR) [ln Ψ()]∇()d ≤ Ψ (⎣ , ) +Ψ ( , ) 0 . ⎦ � − � � 1 for all ∈ [0, 1], such that

1 ≼ ln [Ψ( )×Ψ̃ (�)] h(�)∇((1 − �) + ��)d�. 1 2 1 1 5 2h( ) 25 ∗ + 2h 2 (28) Ψ , ( 2 ) = (2 − )e 2 = (2 − )e 4 , 0 2 If Ψ is log-h-concave FIVF then � 1 1 exp ln Ψ∗(, )d (FR) [ln Ψ()]∇()d − � − ⎛ ⎞ ⎜ 4 ⎟ 1 ⎜ 1 2 ⎟ =⎝exp ln (2 − )ez ⎠ d = eln (2−)+7, ≽ ln [Ψ( )×Ψ̃ (�)] h(�)∇((1 − �) + ��)d�. 3 ⎛ 1 � � ⎞ 0 ⎜ ⎟ ⎜ ⎟ Proof Let Ψ be a log-h-convex FIVF. Then, by Theo- ⎝ 1 ⎠ ∫ h()d ∈ [0, 1], ∗ ∗ rem 2.13, for each we have Ψ ( , ) ×Ψ (, ) 0 1 [ln Ψ∗( +(1 − ), )]∇( +(1 − )) 16 17 = (2 − )e.4(2 − )e 2 = 2(2 − )e 2 . h( ) ln Ψ ( , )+ ≤ ∗ ∇( +(1 − )), h(1 − ) ln Ψ∗(, ) From which, it follows that [ln Ψ∗( +(1 − ), )]∇( +(1 − )) 25 ln(2− )+7 17 (2 − )e 4 ≤ e ≤ 2(2 − )e 2 , h( ) ln Ψ∗( , )+ ≤ ∇( +(1 − )). h(1 − ) ln Ψ∗(, ) (29) that is

25 25 ln( )+7 ln(2− )+7 And e 4 , (2 − )e 4 ≤I e , e 17 17 ln Ψ∗((1 − ) + , ) ∇((1 − ) + ) ≤I 2 e 2 ,2(2 − )e 2 , h(1 − ) ln Ψ ( , )+ ≤ ∗ ∇((1 − ) + ), h( ) ln Ψ (, ) for all ∈ [0, 1]. Hence, ∗ ln Ψ∗((1 − ) + , ) ∇((1 − ) + ) � 1 1 ∗ + � 2h 1 h(1 − ) ln Ψ ( , )+ ( 2 ) Ψ ≼ exp (FR) Ψ( )d ≤ ∗ ∇((1 − ) + ). 2 � − � h( ) ln Ψ (, ) (30) � � ⎡ ⎤ ⎢ 1 ⎥ [ ] ⎢ ∫ h(�)d� ⎥ After adding (29) and (30), and integrating over 0, 1 , ̃ 0 ≼[Ψ( ⎣)×Ψ(�)] . ⎦ we get Now, we discuss second and frst HH-Fejér inequality for log-ℎ-convex FIVF, respectively.

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1 1 ln Ψ ( + (1 − ), ) ∇( + (1 − ))d ln Ψ ( + (1 − ), ) ∇( + (1 − ))d � ∗ ∗ 0 � � 0 1 1 (1 − ) + + ln Ψ∗((1 − ) + , )∇((1 − ) + )d = ln Ψ ((1 − ) + , ) ∇ d � ∗ 0 0 h( )∇( + (1 − )) 1 ln Ψ ( , ) ∗ +h(1 − )∇((1 − ) + ) 1 d , = ln Ψ∗( , ) ∇( )d ≤ � ( − )∇( + ( − )) � − � h 1 1 ⎡ + ln Ψ∗(, ) ⎤ 0 ⎢ +h( )∇((1 − ) + ) ⎥ � � 1 (32) 1 ⎢ ⎥ ⎢ ⎥ ln Ψ∗((1 − ) + , ) ∇((1 − ) + )d ln Ψ⎣∗((1 − ) + , ) ∇((1 − ) + )d ⎦ � 0 0 � � 1 1 + = ln Ψ∗( + (1 − ), ) ∇ d + ln Ψ∗( + (1 − ), )∇( + (1 − ))d (1 − ) � 0 0 h( )∇( + (1 − )) 1 1 ln Ψ∗( , ) = ln Ψ∗(, ) ∇()d. +h(1 − )∇((1 − ) + ) − ≤ � � d . � h(1 − )∇( + (1 − )) ⎡ + ln Ψ∗(, ) ⎤ 0 +h( )∇((1 − ) + ) ⎢ ⎥ From (31) and (32), we have ⎢ � � ⎥ ⎢ ⎥ ⎣ 1 ⎦ 1 ln Ψ (, ) ∇()d − � ∗ = 2 ln Ψ∗( , ) h()∇( + (1 − )) d 1 0 (1 − )u h ≤ ln Ψ∗( , ) ×Ψ∗( , ) ()∇ d, 1 � + 0 + 2 ln Ψ∗(, ) h()∇((1 − ) + ) d, 1 ∗ 0 ln Ψ (, ) ∇()d − � 1 ∗ 1 = 2 ln Ψ ( , ) h()∇( + (1 − )) d (1 − )u ≤ ln Ψ∗( , ) ×Ψ∗( , ) h()∇ d, 0 � + 0 1 + 2 ln Ψ∗(, ) h()∇((1 − ) + ) d. that is 0 1 ∫ ln Ψ∗( , ) ∇( )d , ∇ − Since is symmetric, then ⎡ 1 � ∗ � ⎤ 1 ⎢ ∫ [ln Ψ ( , )]∇( )d ⎥ − Ψ∗( , ) (1 − )u ⎢ ⎥ =2 ln h()∇ d ⎢ ⎥ ×Ψ∗(, ) + 0 ⎢ ⎥ ⎣ ⎦ ∗ 1 (31) ≤I ln[Ψ∗( , )×Ψ∗(, )], ln[Ψ ( , ) ∗ Ψ ( , ) (1 − )u 1 =2 ln ∗ h()∇ d. ×Ψ (, ) + ×Ψ∗(, )]] h()∇((1 − ) + ) d, 0 � 0 Since hence

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� 1 1 ( ) [ Ψ()]∇() 1 + FR ln d ln Ψ∗ , ∇((1 − ) + )d � − 1 2 � h 2 0 1 1 + ̃ ≤ ∫ ln Ψ ( + (1 − ) , ) ∇ d ≼ ln [Ψ( )×Ψ(�)] h(�)∇((1 − �) + ��)d�. ∗ (1 − ) 0 0 1 This concludes the proof. + ln Ψ ((1 − ) + , ) ∇((1 − ) + )d, � ∗ 0 → Theorem 3.5 Let Ψ∶[ , ] 0 be a log −h-convex FIVF 1 → ℝ+ with u <� and h ∶ [0, 1] , for all ∈ [0, 1] , 1 ∗ + + ln Ψ , ∇((1 − ) + )d Ψ ∶ K ⊂ ℝ → ⊂ represent the family of IVFs 1 � C C h 2 � ℱℛ 2 0 through -levels. If Ψ∈ ([�, ],) and ∇∶[ , ] → ℝ, ∇() 0, + , 1 � ≥ symmetric with respect to 2 ∗ + and ∇()d > 0 then ≤ ln Ψ ( + (1 − ) , ) ∇ d , � (1 − ) 0 1 � 2h 1 + � 2 ln Ψ ≼ (FR) [ln Ψ()]∇()d. � (33) + ln Ψ∗((1 − ) + , ) ∇((1 − ) + )d, 2 ∫ ∇( )d � � 0 If Ψ is log-h-concave FIVF then (35) Since 2h 1 � + � 2 1 ln Ψ ≽ (FR) [ln Ψ()]∇()d. � 2 ∫ ∇( )d � ln Ψ ( + (1 − ), ) ∇( + (1 − ))d ∗ 0 Proof Since Ψ is a log-h-convex, then by Theorem 2.13, for 1 ∈ [0, 1], (1 − )u we have = ln Ψ ((1 − ) + , ) ∇ d ∗ + + 1 0 ln Ψ∗ , ≤ ln Ψ∗( + (1 − ) , ) h 1 2 2 1 = ln Ψ (, ) ∇()d, ∗ + ln Ψ∗((1 − ) + , ) − 1 + (34) ln Ψ∗ , ≤ ln Ψ∗( + (1 − ) , ) 1 (36) 1 2 h ∗ 2 ln Ψ ( + (1 − ), ) ∇( + (1 − ))d ∗ + ln Ψ ((1 − ) + , ), 0 1 ∇((1 − ) + ) =∇( + (1 − )) (1 − )u By multiplying (34) by = ln Ψ∗((1 − ) + , ) ∇ d and integrate it by over [0, 1], we obtain + 0 1 = ln Ψ∗(, ) ∇()d, −

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From (35) and (36), we have then, we have 2h 1 + 2 1 ln Ψ , ≤ ln Ψ (, ) ∇()d, ln Ψ∗( , ) ∇( )d ∗ ∗ − 2 ∫ ∇( )d � 8 1 2h 1 ∗ + 2 ∗ = ln Ψ (, ) ∇()d ln Ψ , ≤ ln Ψ (, ) ∇()d. ∗ 7 2 ∫ ∇( )d � 1 9 2 From which, we have 1 = ln Ψ (, ) ∇()d 7 ∗ + ∗ + ln Ψ∗ , , ln Ψ , 1 2 2 8 � � � � �� 1 1 2h ∫ ln Ψ∗( , ) ∇( )d , + ln Ψ∗( , )∇( )d , 2 7 ≤I , 9 �� ⎡ ⎤ 2 ∫ ∇( )d � ∗ � ⎢ ∫ [ln Ψ ( , )]∇( )d ⎥ ⎢ ⎥ 1 ∗ ⎢ ⎥ ln Ψ (, ) ∇()d − that is ⎢ ⎥ ⎣ ⎦ 8 � 2h 1 1 + � 2 = ln Ψ∗(, ) ∇()d ln Ψ ≼ (FR) [ln Ψ()]∇()d, � 7 2 ∫ ∇()d � 1 9 2 1 then we complete the proof. = ln Ψ∗(, ) ∇()d 7 s (37) Remark 3.6 If h( ) = with s ∈ (0, 1), then from Theorems 1 8 3.4 and 3.5, we get result for log-s-convex FIVFs. 1 h( ) = + ln Ψ∗(, ) ∇()d, If then from Theorems 3.4 and 3.5, we obtain 7 result for log-convex FIVFs. 9 Ψ ( , ) =Ψ∗( , ) = 1 2 If ∗ with , then from Theorems 9 3.4 and 3.5, we obtain HH-Fejér inequality for log-ℎ-convex 2 1 function. = [ln (2)]( − 1)d 7 1 Example 3.7 We consider h( ) = , for ∈ [0, 1] and the 8 Ψ∶[ , ] = [1, 8] → 1 FIVFs 0 is given by, + [ln (2)](8 − )d 7 ∈ [ ] 9 , 0, 2 , 2 2 Ψ( )( ) = 4 − , ∈ (2 ,4 ], 7 5 2 = ln (2) + , ⎧ 0, otherwise, 4 2 ⎪ 9 ⎨ 2 ⎪ ∈ [0, 1], 1 Then, for⎩ each we have = [ln (2(2 − ))]( − 1)d 7 Ψ ( ) = [2 ,2(2 − ) ]. 1 8 Ψ ( ) = Ψ∗( ) = ( − ) 1 Since ∗ , 2 z , , 2 2 z log-h-convex + [ln (2(2 − ))](8 − )d ∈ [0, 1] Ψ( ) 7 functions, for each then, by Theorem 2.13 is 9 log-h-convex FIVF. If 2 7 5 = ln (2(2 − )) + , − 1, ∈ 1, 9 4 2 ∇( ) = 2 8 − , ∈ � 9 ,8� , And ⎧ 2 ⎪ ⎨ � � ⎪ ⎩ 1 3 155 Page 12 of 16 International Journal of Computational Intelligence Systems (2021) 14:155

1 From (39) and (40), we have Ψ∗( , ) ln h()∇(u + (, u)) d [ln (9 ), ln (9(2 − ))] ×Ψ∗(, ) � � 0 ≤ [ln (2 ) + 1.54, ln (2(2 − )) + 1.54]. 1 I Ψ∗( , ) ln h()∇(u + (, u)) d ×Ψ∗(, ) Hence, Theorem 3.5 is verifed. � � 0 1 h 2 1 Now, we prove the Jensen’s inequality for log- -convex FIVF. = ln 32 2 72d + (7 − 7)d ⎡ ⎤ (38) 0 1 Theorem 3.8 Let ∈ ℝ+ u ∈ [ , ], � � ��⎢ 2 ⎥ j , j → 7 ⎢ ⎥ (j = 1, 2, 3, … k, k ≥ 2) and Ψ∶[ , ] 0 be a log h = ln 32 2 ⎢ ⎥ 8 ⎣ ⎦ -convex FIVF with non-negative real valued function → ℝ ℝ → + 1 h ∶ [0, 1] , for all ∈ [0, 1] , Ψ� ∶ K ⊂ rep- � � �� 2 1 C resent the family of IVFs through -levels. If h is multiplica- = ln 32(2 − )2 72d + (7 − 7)d tive function then, ⎡ 0 1 ⎤ � � ��⎢ 2 ⎥ k k � h j ⎢ ⎥ 1 W 7 2 Ψ � z ≼ Ψ z k , = ln 32(2 − ) ⎢ ⎥ W j j j (41) 8 ⎣ ⎦ k j=1 j=1 � � �� From (37) and (38), we have If Ψ is log-h-concave FIVF then

7 5 7 5 k k � ln (2 ) + , ln (2(2 − )) + 1 h j 4 2 4 2 Ψ � z ≽ Ψ z Wk , W j j j 7 2 7 2 k j=1 j=1 ≤I ln 32 , ln 32(2 − ) , 8 8 W = k . Ψ h where k j=1 j If is log- -concave then, (41) is for all ∈ (0, 1]. reversed. ∑ Hence, Theorem 3.4 is verifed. For Theorem 3.5, we have Proof When k = 2 , (41) is true. Consider (16) is true for + k = n − 1, then by Theorem 2.13, we have ln Ψ∗ , = ln (9), 2 n−1 n−1 � (39) 1 h j ∗ + Ψ � z ≼ Ψ z Wn−1 , ln Ψ , = ln (9(2 − )), W j j j 2 n−1 j=1 j=1 9 Now, let us prove that (41) holds for k = n. 2 8 49 n ∇()d = ( − 1)d + (8 − )d = , 1 4 Ψ jzj 1 9 W � n j=1 � 2 � n−2 + 1 z + n−1 n n−1 z 1 W j j W + n−1 2h =Ψ n−2 j=1 n n−1 n . 2 ln Ψ (, ) ∇()d ⎛ + n z� � ⎞ ∗ ∑ + n ∇ ()d � n−1 n ∫ ⎜ ⎟ ⎜ ⎟ = ln (2) + 1.54 ⎜ ∈ [0, 1] ⎟ Therefore,⎝ for every , we have ⎠ (40) 2h 1 2 ln Ψ∗(, ) ∇()d ∇ ()d � ∫ = ln (2(2 − )) + 1.54

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n and the result follows. 1 Ψ∗ jzj, Wn ⋯ � j=1 � If 1 = 2 = 3 = = k = 1, then Theorem 3.8 reduces �n 1 to the following: Ψ∗ z , W j j � n j=1 � j = 1, 2, 3, … k, � Corollary 3.9 Let u ∈ [ , ], and n−2 j + k ≥ 2 1 n−1 n n−1 jzj + zn−1 → Wn Wn n−1+ n Ψ∶[ , ] 0 be a log-h-convex FIVF with non-negative ≤ Ψ∗ j=1 , → ℝ ⎛ + n z�, � ⎞ real valued function h ∶ [0, 1] . For all ∈ [0, 1] , ∑ + n ⎜ n−1 n ⎟ ℝ → + Ψ� ∶ K ⊂ C ⊂ C represent the family of IVFs ⎜ n−2 ⎟ + through -levels. If h is multiplicative function, then ⎜ 1 z + n−1 n n−1 z ⎟ ∗⎝ W j j W + n−1 ⎠ ≤ Ψ n j=1 n n−1 n , k k ⎛ + n z�, � ⎞ 1 h 1 ∑ + n Ψ z ≼ Ψ z k , ⎜ n−1 n ⎟ k j j (42) j=1 j=1 n−2⎜ ⎟ h j ⎜ W ⎟ ≤ Ψ∗ z , n ⎝ j � � ⎠ Ψ h j=1 If is a log- -concave, then (42) is reversed. � � � �� + h n−1 n n−1 Wn ℝ+ zn−1 Theorem 3.10 Let j ∈ uj ∈ [ , ], n−1+ n � � , × Ψ∗ n , → + z , (j = 1, 2, 3, … k, k ≥ 2) and Ψ∶[ , ] 0 be a log h � � + n �� - n−1 n convex FIVF with non-negative real valued function n−2 - j + ∗ h h ∶ [0, 1] → ℝ ∈ [0, 1] Ψ ∶ K ⊂ ℝ → ⊂ ≤ Ψ z , Wn . For all , � C C j � � j=1 represent the family of IVFs through -levels. If � � � �� − + n (L, U) ⊆ [ , �]h is multiplicative function then, h n 1 n−1 Wn zn−1 ∗ n−1+ n � � k k × Ψ �j U−zj �j zj−L �j n , h h h h h + z , W U−L W M−L W + n Ψ z k ≼ [Ψ(L)] k × [Ψ(U)] k , � � n−1 n �� j j=1 j=1 n−2 h j h n−1 W W ≤ Ψ∗ z , n × Ψ∗ z − , n (43) j � � n 1 � � j=1 If Ψ is log-h-concave FIVF, then � � � �� h n Wn k � k U−z � z −L � × Ψ∗ zn, , h j h j h j h j h j � � W U−L W M−L W Ψ zj k ≽ [Ψ(L)] k × [Ψ(U)] k , n−2 h j h n−1 j=1 j=1 � �∗ �� W ∗ W ≤ Ψ z , n × Ψ z − , n j � � n 1 � � j=1 W = k . Ψ h � where k j=1 j If is log- -concave, then (43) is � � �� n � � �� ∗ h × Ψ z , Wn , reversed. n � � ∑ n h j � � �� W Proof = z , z = z , (j = 1, 2, 3, … k) U = z = Ψ∗ z , n , Consider 1 j 2 , 3 in j � � j=1 (43). Then, for each ∈ [0, 1] , then by Theorem 2.13, we � n � � �� h j have ∗ W = Ψ zj, n , � � U−z z −L h j h j j=1 U−L M−L Ψ∗ z , ≤ Ψ∗(L, ) × Ψ∗(U, ) , � � � �� j U−zj zj−L From which, we have ∗ ∗ h ∗ h Ψ z , ≤ Ψ (L, ) U−L × Ψ (U, ) M−L . j n n 1 1 Ψ z , , Ψ∗ z , ∗ W j j W j j Above inequality can be written as, n j=1 n j=1 U−z h j h j h j n n W U−L W h j h j Ψ∗ zj, k ≤ Ψ∗(L, ) k Wn ∗ Wn ≤I Ψ∗ zj, , Ψ zj, , z −L h j h j j=1 j=1 × Ψ(U, ) M−L Wk ∗ j U−zj j (44) ∗ h ∗ h h that is, Ψ z, Wk ≤ Ψ (L, ) U−L Wk j n n zj−L j �j h h h ∗ M−L W 1 W × Ψ(U, ) k . Ψ � z ≼ Ψ z n , W j j j n j=1 j=1

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k � Taking multiplication of all inequalities (44) for h j j = 1, 2, 3, … k, Ψ z Wk we have j j=1 k j k h U−zj �j zj−L �j Ψ z , Wk h h h h ∗ j � � ≼ [Ψ(L)] U−L Wk × [Ψ(U)] M−L Wk , j=1 j=1 � � � U−z k h j h j U−L W Ψ∗(L, ) k ≤ � z −L� � � , this completes the proof. h j h j M−L W j=1 ×� Ψ∗(U, �) k � ⎛ � � � � ⎞ ∗ k Remark 3.1.1 If Ψ∗( , ) =Ψ (, ) with = 1 , then from ⎜ j ⎟ ∗ ⎜ �h � ⎟ h Ψ z , Wk Theorem 3.8 and Theorem 3, we obtain result for -convex j⎝ � � ⎠ j=1 function, see [23]. ∗ s � � � U−z Ψ ( , ) =Ψ (, ) = 1 h( ) = k h j h j If ∗ with and , then ∗ U−L W [Ψ (L, )] k from Theorem 3.8 and Theorem 3, we obtain result for s ≤ � zj−L� � �j . ∗ h h j=1 ×[Ψ (U, )] M−L Wk -convex FIVF, see [43]. � ⎛ � � � � ⎞ Ψ ( , ) =Ψ∗(, ) = 1 h( ) = ⎜ ⎟ If ∗ with and , then from ⎜ ⎟ Theorem 3.8 and Theorem 3, we obtain result for convex that is ⎝ ⎠ FIVF, see [23]. k h j Ψ z Wk j � � j=1 � � � k k h j h j W ∗ W 4 Conclusion and Future Plan = Ψ∗ zj, k , Ψ zj, k � j=1 � � j=1 � � � ∏ � � ∏U−z � � h j h j Hermite–Hadamard and Jensen’s inequalities are hold for k U−L W Ψ∗(L, ) k this new class of convex FIVFs. Moreover, we have dis- � z −L � � � , h j h j j= M−L W 1 Ψ (U, ) k cussed some special cases which can be obtained by main ⎡ ⎛ � ∗ � � � � � ⎞ ⎤ ≤I ∏ U−zj j , results. In future, we intend to discuss generalized log-h ⎢ k ⎜ ∗ h h ⎟ ⎥ �[Ψ (L, )]� U−L Wk -convex functions. We hope that this notion will assist ⎢ ⎜ � z −L � � � ⎟ ⎥ ⎝ h j h j ⎠ ⎢ j= ∗ M−L W ⎥ 1 [Ψ (U, )] k other authors in fulflling their tasks in various disciplines ⎢ ⎛ � � � � ⎞ ⎥ ∏ U−z of study. ⎢ ⎜ h j h j ⎟ ⎥ k U−L W ⎢ ⎜ Ψ∗(L, ) k ⎟ ⎥ ⎣ ⎝ � z −L � � � ⎠, ⎦ Acknowledgements h j h j The authors would like to thank the Rector, j=1 M−L W �Ψ∗(U, )� k COMSATS University Islamabad, Islamabad, Pakistan, for providing ⎡ ⎛ � � � � ⎞ ⎤, ≤I ∏ U−zj j excellent research and academic environments. ⎢ k ⎜ ∗ h h ⎟ ⎥ �[Ψ (L, )]� U−L Wk ⎢ ⎜ � z −L � � � ⎟ ⎥ ⎝ h j h j ⎠ Authors’ contributions ⎢ j= ∗ M−L W ⎥ All authors contributed equally to the writing of 1 [Ψ (U, )] k ⎢ ⎛ � � � � ⎞ ⎥ this paper. All authors read and approved the fnal manuscript. ∏ U−z ⎢k ⎜ h j h j ⎟ ⎥ U−L W ⎢ ⎜ Ψ∗(L, ) k ,⎟ ⎥ Funding ≤ ⎣ ⎝ � U−z � � � ⎠ ⎦ This study was funded by Fundamental Research Grant I h j h j ∗ U−L W Scheme, FRGS/1/2018/STG06/UMT/01/1, Vote No. 59522, Ministry j=1 �[Ψ (L, )�] k � ⎛⎡ � � � � ⎤⎞ of Higher Education Malaysia, and University Malaysia Terengganu. z −L k ⎜⎢ h j h j ⎥⎟ ⎜⎢ M−L Wk ⎥⎟ Ψ∗(U, ) , Declarations × ⎝⎣ �zj−L � �j � ⎦⎠ ∗ h h j=1 � [Ψ (U)]� M−L Wk � ⎛⎡ � � � � ⎤⎞ Confict of interest The authors declare that they have no competing k ⎜⎢ U−z ⎥⎟ interests. ⎜⎢ h j h j ⎥⎟ = Ψ (L) U−L Wk ⎝⎣ � � � � ⎦⎠ j=1 � � � Open Access This article is licensed under a Creative Commons Attri- k z −L h j h j bution 4.0 International License, which permits use, sharing, adapta- × Ψ (U) M−L Wk . tion, distribution and reproduction in any medium or format, as long � � � � j=1 as you give appropriate credit to the original author(s) and the source, � � � provide a link to the Creative Commons licence, and indicate if changes Thus, were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated

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