# Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral

Authors: Hu Ge-JiLe; Saima Rashid; Fozia Bashir Farooq; Sobia Sultana
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonicallyψ-s-convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically ψ-s-convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.


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## Body

## 1. Introduction

Fractional calculus is a developing arena in mathematics with profound presentations in all connected areas such as wave liquids, thermodynamics, image processing, virology, biological population models, chaos, and signals processing, see [1–5]. It has been employed to model physical phenomena that are originated to be paramountly described by fractional differential equations. Nowadays, the local fractional calculus is trying to report the nondifferentiable problems, for example, heat conduction problem involving local derivatives of fractional order [6, 7], local fractional Tricomi equation [8], and fractal vehicular traffic flow [9].Also, several noted variants via local fractional integral have been investigated by many researchers, for example, Mo et al. [10] propounded Hermite-Hadamard inequalities for generalized s-convex functions. Abdeljawad et al. [11] introduced some local fractional variants for generalized s,m-convex functions in the fractal style with applications. Also, Sun and Liu [12] reported new generalizations of Hermite-Hadamard inequalities for generalized harmonically convex functions. For further investigations concerning to local fractional integral inequalities, we recommend the readers to [13–15] and the references therein.It is well known that convexity theory has potential applications in many intriguing and captivating fields of research and furthermore played a remarkable role in numerous areas, such as coding theory, optimization, physics, information theory, engineering, and inequality theory. Several new classes of classical convexity have been proposed in the literature (see [16–26]). Many researchers endeavored, attempted, and maintain their work on the concept of convex functions, generalize its variant forms in different ways using innovative ideas and fruitful techniques [27–29]. In this regard, integral inequalities have played an important role in describing real world problems such as in elementary research conducted by Niculescu and Persson [30]. In this context, Hermite-Hadamard [31, 32] like type inequalities are very influential in convexity theory, which describes that if Q:I↦ℝ is convex function, then the double inequality
(1)Qa1+a22≤1a2−a1∫a1a2Qzdz≤Qa1+Qa22,holds for all a1,a2∈I with a1≠a2.Numerous approaches have been done by several researchers to inquire novel generalizations, speculations, and modifications for the Hermite-Hadamard inequality and its variant forms; we prescribe the related phenomena [33–36] to interested readers.Now, we evoke the concept of harmonically convex function [37] as follows.Definition 1 ([37]).
Letℍ⊂ℝ be an interval, and a real-valued function Q:ℍ↦ℝ is said to be harmonically convex if the inequality
(2)Qa1a2t1a1+1−t1a2≤1−t1Qa1+t1Qa2,holds for all a1,a2∈ℍ and t1∈0,1.In [37], İscan derived the Hermite-Hadamard inequality for harmonically convex function.
(3)Q2a1a2a1+a2≤a1a2a2−a1∫a1a2Qzz2dz≤Qa1+Qa22,holds for all a1,a2∈ℍ with a1<a2 and Q∈L1a1,a2 (the set of Lebesgue integrable functions).To make this collection more comprehensive, we introduce the notion of generalized harmonicallyψ-s-convex functions in the sense of Raina’s function on fractal set. By considering generalized harmonically ψ-s-convex functions, certain refinements of Hermite-Hadamard type inequalities are derived that can be used to characterize some useful consequences of harmonically ψ-convex functions with the appropriate assumption of α^, which are connected to Raina’s function. Remarkable special cases are captured, and new Ostrowski type results are proved for the aforesaid class of functions. Additionally, our findings for the new schemes are also constituted some better approaches in various directions for proposing variation in α^, accordingly Remark 12. Application of the α^-type special mean and Mittag-Leffler functions are presented by mingling the suggested methodology and the theory of the special functions. We hope that the idea and results obtained herein will be a catalyst for further investigation.


## 2. Prelude

In what follows, we demonstrate a different notion of differentiation that merges the idea of fractional differentiation and fractal derivative; recently, one has established the accompanying sets [6].For the sake of brevity, we mentionℕ (the set of natural numbers), ℤ (set of positive integers), ℚ (set of rational number), and ℝ (set of real numbers), respectively.Also, whenever theα^-type set ℝα^ of real line numbers are included, the α^ is supposed to be implicitly 0<α^≤1. One has also characterized by two binary operations the addition “+” and the multiplication “.” (which is conventionally omitted) on the α^-type set ℝα^ of real line numbers as follows.Fore1α^,e2α^∈ℝα^, two binary operations the addition “+” and the multiplication “.” are defined as e1α^+e2α^≔e1+e2α^ and e1α^.e2α^≔e1.e2α^, for e1α^,e2α^∈ℝα^.Then, we have the following asserations to show(1)
ℝα^,+ is an abelian group: for e1α^,e2α^,e3α^∈ℝα^(i)
e1α^+e2α^∈ℝα^(ii)
e1α^+e2α^=e2α^+e1α^(iii)
e1α^+e2α^+e3α^=e1α^+e2α^+e3α^(iv)
0α^+e1α^=e1α^+0α^=e1α^;0α^ is the additive identity for ℝα^,+(v)
e1α^+−e1α^=−e1α^+e1α^=0α^; for any eα^∈ℝα^,−eα^ is the inverse element of eα^ for ℝα^,+(2)
ℝα^\0α^,. is an abelian group: for e1α^,e2α^,e3α^∈ℝα^(vi)
e1α^.e2α^∈ℝα^(vii)
e1α^.e2α^=e2α^.e1α^(viii)
e1α^.e2α^.e3α^=e1α^.e2α^.e3α^(ix)
1α^+e1α^=e1α^+1α^=e1α^;1α^ is the multiplicative identity for ℝα^,.(x)
e1α^1/e1α^=1/e1α^e1α^=1α^; for any eα^∈ℝα^\0α^,1/eα^ is the inverse element of eα^ for ℝα^,.(3)
Distributive law holds:e1α^e2α^+e3α^=e1α^e2α^+e1α^e3α^Proposition 2.
The following asserations holds true:(i)
ℝα^,+,. is a field(ii)
The additive identity0α^ and the multiplication identity 1α^ are unique(iii)
The additive inverse element and the multiplicative inverse element are unique(iv)
For eache1α^∈ℝα^\0α^, its inverse element 1/e1α^ may be written as 1α^/e1α^ but not as 1/e1α^; for each e1α^∈ℝα^, its inverse element −e1α^ may be written as −e1α^(v)
If the order< is defined on ℝα^,+ as follows: e1α^<e2α^∈ℝα^ if and only if e1<e2∈ℝ, then ℝα^,+,.,< is an ordered field like ℝ,+,.,<In [6], Yang introduced the following notions for local fractional continuity and differentiability as follows:Definition 3 ([6]).
A nondifferentiable mappingQ:ℝ→ℝα^,t1→Qε is said to be local fractional continuous at ε∘, if for any ε>0, there exists m>0, satisfying that
(4)Qε−Qε∘<εα^,holds for ∣ε−ε∘∣<m. If Qε is local continuous on a1,a2, then we symbolize it by Qε∈ℂα^a1,a2.Definition 4 ([6]).
The local fractional derivative ofQε of order α^ at ε=ε∘ is defined by the expression
(5)Qα^ε∘=ε∘Dεα^Qε=dα^Qεdεα^ε=ε∘=limε→ε∘Δα^Qε−Qε∘ε−ε∘α^,where Δα^Qε−Qε∘=Γα^+1Qε−Qε∘. Let Qα^ε=Dεα^Qε. If there exists Qk+1α^ε=Dεα^⋯Dεα^⏞k+1timesQε for any ε∈Ω⊆ℝ, then it is denoted by Q∈Dk+1α^I, where k=0,1,2,⋯.Definition 5.
LetQε∈ℂα^a1,a2, and let Δ=ρ0,ρ1,⋯,ρN,N∈ℕ be a partition of a1,a2 which satisfies a1=ρ0<ρ1<⋯<ρN=a2. Then, the local fractional integral of Q on a1,a2 of order α^ is defined as follows
(6)a1Ia2α^Qε=1Γ1+α^∫a1a2Qρdρα^≔1Γ1+α^limδρ→0∑j=0N−1QρjΔρj,where δρ≔maxΔρ1,Δρ2,⋯,ΔρN−1 and Δρj≔ρj+1−ρj,j=0,⋯,N−1.
Here, it follows thata1Ia2α^Qε=0 if a1=a2 and a1Ia2α^Qε=−a2Ia1α^Qε if a1<a2. For any ε∈a1,a2, if there exists a1Ia2α^Qε, then it is denoted by Qε∈Iεα^a1,a2.Lemma 6 ([6]).
(1)
Suppose thatQz=Qα^z∈ℂα^a1,a2, then
(7)a1Ia2α^Qz=Qa2−Qa1.(2)
Suppose thatQz,Wz∈Dα^a1,a2, and Qα^z,Wα^z∈ℂα^a1,a2, then
(8)a1Ia2α^QzWα^z=QuWza1a2−a1Ia2α^Qα^zWz.Lemma 7 ([6]).
(9)dα^zkα^dzα^=Γ1+kα^Γ1+k−1α^zk−1α^,1Γ1+α^∫a1a2zkα^dzα^=Γ1+kα^Γ1+k+1α^a2k+1α^−a1k+1α^,k>0.An analog in the fractal setℝα^ of the classical Hölder’s inequality has been established by [15], which is asserted by the following lemma.Lemma 8 ([15].
Generalized Hölder’s inequality). Forω1,ω2>1 with ω1−1+ω2−1=1, and let Q,W∈ℂα^a1,a2, then
(10)1Γ1+α^∫a1a2∣QzWz∣dzα^≤1Γ1+α^∫a1a2Qzω1dzα^1/ω11Γ1+α^∫a1a2Wzω2dzα^1/ω2.In the work of Sun and Liu [13], we present the following concept related with the notion of generalized harmonically convexity.Definition 9 ([13]).
Letℍ⊂0,∞ be an interval and let then Q:ℍ↦ℝα^0<α^≤1 is said to be generalized harmonically convex function if the inequality holds:
(11)Qa1a2t1a1+1−t1a2≤t1α^Qa1+1−t1α^Qa2,∀a1,a2∈ℍ,t1∈0,1.Recently, several noted consequences and motivating facts related to fractal theory have been appeared in the literature [5, 14, 15]. In [13], Sun explored the following analog of the Hermite-Hadamard inequality (3) for generalized harmonically convex functions.
(12)1Γ1+α^Q2a1a2a1+a2≤a1a2a2−a1α^a1Ia2α^Qzz2α^≤Γ1+α^Γ1+2α^Qa1+Qa2.Firstly, letλ=λmm=0∞ be a bounded sequence of real numbers, and Fρ,σλ.ρ,σ>0 denotes the Raina’s function.In [38], Raina explored a new class of functions stated as:
(13)Fρ,σλz=Fρ,σλ0,λ1,⋯z=∑m=0∞λmΓρm+λzm,where ρ,σ>0,∣z∣<ℝ and
(14)λ=λ0,⋯.,λm,⋯,is bounded sequence of positive real numbers. Notice that if we choose ρ=1,σ=0 in (13), then
(15)λm=δ1mδ2mδ3mform=0,1,2,⋯,where δ1,δ2, and δ3 are parameters which can choose aribtrary real and complex values (provided that δ3≠0,−1,−2,⋯,), and we have the notion bm by
(16)bm=Γb+mΓb=bb+1⋯b+m−1,m=0,1,2,⋯,then the classical hypergeometric function stated as follows
(17)Fρ,σλz=Fδ1,δ2;δ3;z=∑m=0∞δ1mδ2mm!δ3mzm,∣z∣≤1,z∈ℂ.Also, ifλ=1,1,⋯ with ς=δ,Rδ>0,λ=1 and restricting its domain to z∈ℂ in (13), then we have the classical Mittag-Leffler function:
(18)Eδ1z=∑m=0∞1Γ1+δ1mzm.Next, we evoke a novel concept of harmonic set and mappings including Raina’s functions.Definition 10.
Letρ,σ>0 and λ=λmm=0∞ be a bounded sequence of real numbers. A nonempty set ℍF⊂0,∞ is said to be generalized harmonically ψ-convex set, if
(19)a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1∈ℍF,for all a1,a2∈ℍF,t1∈0,1.Next, we introduce the notion of generalized harmonicallyψ-s-convex function as follows:Definition 11.
Letρ,σ>0 and λ=λmm=0∞ be a bounded sequence of real numbers. If a mapping Q:ℍF↦ℝα^0≤α^<1 satisfies the following inequality:
(20)Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1≤t1sα^Qa1+1−t1sα^Qa2,holds for all a1,a2∈ℍF,t1∈0,1 and s∈0,1.Remark 12.
In view of Definition11, we have that
(i)
Choosingα^=1, then we acquire harmonically ψ-convex function as follows:
(21)Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1≤t1sQa1+1−t1sQa2,∀a1,a2∈ℍF,t1∈0,1,s∈0,1.(ii)
If we takes=1 and Fρ,σλa2−a1=a2−a1, then we get Definition 3.1 in [13](iii)
If we takeFρ,σλa2−a1=a2−a1 along with α^=s=1, then we get Definition in [37]Recall the two special functions on Yang’s fractal sets.(1)
Generalized beta function(22)Bα^z1,z2=1Γ1+α^∫01t1α^z1−11−t1α^z2−1dt1α^,z1,z2>0.(2)
Generalized hypergeometric function(23)2F¯1α^z1;z2;z3,z=1Bα^z2,z3−z21Γ1+α^∫01t1α^z1−11−t1α^z3−z2−11−zt1−α^z1dt1α^,0<z2<z3,∣z∣<1.


## 3. Hermite-Hadamard Type Inequality

In this section, we derive a novel version of Hermite-Hadamard inequalities for generalized harmonicallyψ-convex functions, and this is the main motivation of this paper.Theorem 13.
Fors∈0,1,ρ,σ>0,λ=λmm=0∞ a bounded sequence of real numbers and Q:IF=a1,a1+Fρ,σλa2−a1⊂ℝ\0↦ℝα^0<α^≤1 be generalized harmonically ψ-s-convex function where a1,a1+Fρ,σλa2−a1∈IF,Fρ,σλa2−a1>0 such that Q∈Dα^a1,a1+Fρ,σλa2−a1 and Qα^∈ℂα^a1,a1+Fρ,σλa2−a1. Then, the following inequalities hold:
(24)2s−1α^Γ1+α^Q2a1a1+Fρ,σλa2−a12a1+Fρ,σλa2−a1≤a1α^a1+Fρ,σλa2−a1α^Fρ,σλa2−a1α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^≤Γ1+sα^Γ1+s+1α^Qa1+Qa2.Proof.
SinceQ is generalized harmonically ψ-s-convex function with t1=1/2, then
(25)Q2xx+Fρ,σλy−x2x+Fρ,σλy−x≤Qx+Qy2sα^,∀x,y∈IF,t1∈0,1.
Letx=a1a1+Fρ,σλa2−a1/a1+1−t1Fρ,σλa2−a1 and y=a1a1+Fρ,σλa2−a1/a1+t1Fρ,σλa2−a1 in (25). Then, by local fractional integration over 0,1, we have
(26)2sα^Γ1+α^∫01Q2a1a1+Fρ,σλa2−a12a1+Fρ,σλa2−a1dt1α^≤1Γ1+α^∫01Qa1a1+Fρ,σλa2−a1a1+1−t1Fρ,σλa2−a1+Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^=2α^a1α^a1+Fρ,σλa2−a1α^Fρ,σλa2−a1α^1Γ1+α^∫a1a1+Fρ,σλa2−a1Qzz2α^dzα^=2α^a1α^a1+Fρ,σλa2−a1α^Fρ,σλa2−a1α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^.
Also, using the fact that(27)2sα^Γ1+α^∫01Q2a1a1+Fρ,σλa2−a12a1+Fρ,σλa2−a1dt1α^=2sα^Γ1+α^Q2a1a1+Fρ,σλa2−a12a1+Fρ,σλa2−a1,we have
(28)2s−1α^Γ1+α^Q2a1a1+Fρ,σλa2−a12a1+Fρ,σλa2−a1≤a1α^a1+Fρ,σλa2−a1α^Fρ,σλa2−a1α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^.
For the proof of other inequality in (73), noting that Q is a generalized harmonically ψ-s-convex mapping, for t1∈0,1, we have
(29)Qa1a1+Fρ,σλa2−a1a1+1−t1Fρ,σλa2−a1+Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1≤t1sα^+1−t1sα^Qa1+Qa2.
Again, using local fractional integration for above inequality with respect tot1 over 0,1 yields
(30)a1α^a1+Fρ,σλa2−a1α^Fρ,σλa2−a1α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^≤Γ1+sα^Γ1+s+1α^Qa1+Qa2.
Combining (28) and (30), we get the inequality (24). Hence, the proof is completed.Certain noteworthy cases of Theorem13 are established as follows:
(i)
Choosingα^=1, then we attain a new consequence for harmonically ψ-s-convex functionCorollary 14.
Forρ,σ>0,λ=λmm=0∞ a bounded sequence of real numbers and Q:IF=a1,a1+Fρ,σλa2−a1⊂ℝ\0↦ℝ be harmonically ψ-s-convex function where a1,a1+Fρ,σλa2−a1∈IF,Fρ,σλa2−a1>0 such that Q′∈L1a1,a1+Fρ,σλa2−a1. Then, the following inequalities hold:
(31)2s−1Q2a1a1+Fρ,σλa2−a12a1+Fρ,σλa2−a1≤a1a1+Fρ,σλa2−a1Fρ,σλa2−a1∫a1a1+Fρ,σλa2−a1Qzz2dz≤Qa1+Qa2s+1.Remark 15.
In view of Theorem13, we have that
(i)
Lettings=1 and Fρ,σλa2−a1=a2−a1, then we get Theorem 4.1 of [13].(ii)
Lettingα^=1 and Fρ,σλa2−a1=a2−a1, then we get Theorem 3 of [39].Lemma 16.
Forρ,σ>0,λ=λmm=0∞ a bounded sequence of real numbers and Q:IF∘=a1,a1+Fρ,σλa2−a1⊂ℝ\0↦ℝα^0<α^≤1 (IF∘ is the interior of IF)be a generalized harmonically ψ-convex function, where a1,a1+Fρ,σλa2−a1∈IF∘,Fρ,σλa2−a1>0 such that Q∈Dα^a1,a1+Fρ,σλa2−a1 and Qα^∈ℂα^a1,a1+Fρ,σλa2−a1. Then, the identity
(32)ΩQa1,a1+Fρ,σλa2−a1;ω,α^≔Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^1Γ1+α^∫01−ωt1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12α^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^+1Γ1+α^∫1−ω1t1−1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12α^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^,where
(33)ΩQa1,a1+Fρ,σλa2−a1;ω,α^≔1−ωα^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1+ωα^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1−a1a1+Fρ,σλa2−a1Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^,holds for ω∈0,1 and α^∈0,1.Proof.
By local fractional integration by parts, we have(34)J1=1Γ1+α^∫01−ωt1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12α^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^=t1α^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a101−ω−Γ1+α^Γ1+α^∫01−ωQa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^=1−ωα^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1−Γ1+α^Γ1+α^∫01−ωQa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^.
Settingz=a1a1+Fρ,σλa2−a1/a1+t1Fρ,σλa2−a1, then
(35)J1=1−ωα^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1−a1a1+Fρ,σλa2−a1Fρ,σλa2−a1α^Γ1+α^Γ1+α^∫a1a1a1+Fρ,σλa2−a1a1+ωFρ,σλa2−a1Qzz2α^dzα^=1−ωα^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1−a1a1+Fρ,σλa2−a1Fρ,σλa2−a1α^Γ1+α^a1Ia1a1+Fρ,σλa2−a1/a1+ωFρ,σλa2−a1α^Qzz2α^.
Analogously, we have(36)J2=1Γ1+α^∫1−ω1t1−1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12α^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^=ωα^Qa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1−a1a1+Fρ,σλa2−a1Fρ,σλa2−a1α^Γ1+α^a1a1+Fρ,σλa2−a1/a1+ωFρ,σλa2−a1Ia1+Fρ,σλa2−a1α^Qzz2α^.
AddingJ1 and J2, we obtain the required result. This completes the proof of Lemma 16.Theorem 17.
Assume that the hypothesis of Lemma16 are satisfied. If Qα^q1 is a generalized harmonically ψ-s-convex function on IF for p,q>1,p−1+q−1=1. Then, the following inequality for local fractional integrals holds
(37)ΩQa1,a1+Fρ,σλa2−a1;ω,α^≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+α^Γ1+2α^1−1/q×1−ω21−1/qα^Λ1α^q,ω,a1,a2;sQα^a1q+Λ2α^q,ω,a1,a2;sQα^a2q1/q+ω21−1/qα^Λ3α^q,ω,a1,a2;sQα^a1q+Λ4α^q,ω,a1,a2;sQα^a2q1/q,where
(38)Λ1α^q,ω,a1,a2;s≔1−ωs+2α^a1+Fρ,σλa2−a12qα^×Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;ω−1Fρ,σλa2−a1a1,Λ2α^q,ω,a1,a2;s≔1−ω2α^a1+Fρ,σλa2−a12qα^Γ1+α^Γ1+2α^×2F¯1α^2q,2;3;ω−1Fρ,σλa2−a1a1−Λ1α^q,ω,a1,a2;s,Λ3α^q,ω,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+sα^Γ1+s+1α^2F¯1α^2q,s+1;s+2;ω−1Fρ,σλa2−a1a1−Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;ω−1Fρ,σλa2−a1a1−1−ωs+1α^Γ1+sα^Γ1+s+1α^×2F¯1α^2q,s+1;s+2;ω−1Fρ,σλa2−a1a1+Λ1α^q,ω,a1,a2;s,Λ4α^q,ω,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,1;s+3;ω−1Fρ,σλa2−a1a1−1−ωα^Γ1+α^2F¯1α^2q,1;2;ω−1Fρ,σλa2−a1a1+1−ωs+2α^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;ω−1Fρ,σλa2−a1a1.Proof.
By means of Lemma16, the generalized power mean inequality and the generalized harmonically ψ-s-convexity of Qα^q, we have
(39)∣ΩQa1,a1+Fρ,σλa2−a1;ω,α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^1Γ1+α^∫01−ωt1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12α^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1dt1α^+1Γ1+α^∫1−ω1t1−1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12α^∣Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1∣dt1α^≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^1Γ1+α^∫01−ωt1α^dt1α^1−1/q×1Γ1+α^∫01−ωt1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1qdt1α^1/q+1Γ1+α^∫1−ω11−t1α^dt1α^1−1/q×1Γ1+α^∫1−ω11−t1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1qdt1α^1/q≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^1Γ1+α^∫01−ωt1α^dt1α^1−1/q×1Γ1+α^∫01−ωa1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^t1s+1α^Qα^a1q+t1α^1−t1sα^Qα^a2qdt1α^1/q+1Γ1+α^∫1−ω11−t1α^dt1α^1−1/q×1Γ1+α^∫1−ω1a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^1−t1α^t1sα^Qα^a1q+1−t1s+1α^Qα^a2qdt1α^1/q.
Employing Lemma7, we have that
(40)1Γ1+α^∫01−ωt1α^dt1α^=Γ1+α^Γ1+2α^1−ω2α^,1Γ1+α^∫1−ω11−t1α^dt1α^=Γ1+α^Γ1+2α^ω2α^.
Settingt1=1−ωη, with the aid of Lemma 7 and simple computations yield
(41)Λ1α^q,ω,a1,a2;s≔1Γ1+α^∫01−ωt1s+1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^dt1α^=a1a1+Fρ,σλa2−a12qα^Γ1+α^∫01−ωt1s+1α^a1+t1Fρ,σλa2−a1−2qα^dt1α^=1−ωs+2α^a1+Fρ,σλa2−a12qα^Γ1+α^∫01t1s+1α^1−ηω−1Fρ,σλa2−a1a1−2qα^dηα^=1−ωs+2α^a1+Fρ,σλa2−a12qα^Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;ω−1Fρ,σλa2−a1a1.
In a similar way, we can find(42)Λ2α^q,ω,a1,a2;s≔1Γ1+α^∫01−ωt1α^1−t1sα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^dt1α^=a1a1+Fρ,σλa2−a12qα^Γ1+α^∫01−ωt1α^a1+t1Fρ,σλa2−a1−2qα^−t1s+1α^a1+t1Fρ,σλa2−a1−2qα^dt1α^=1−ω2α^a1+Fρ,σλa2−a12qα^Γ1+α^Γ1+2α^2F¯1α^2q,2;3;ω−1Fρ,σλa2−a1a1−1−ωs+2α^a1+Fρ,σλa2−a12qα^Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;1−ω−1Fρ,σλa2−a1a1=1−ω2α^a1+Fρ,σλa2−a12qα^Γ1+α^Γ1+2α^2F¯1α^2q,2;3;ω−1Fρ,σλa2−a1a1−Λ1α^q,ω,a1,a2;s,(43)Λ3α^q,ω,a1,a2;s≔1Γ1+α^∫1−ω1t1sα^1−t1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^dt1α^=a1a1+Fρ,σλa2−a12qα^Γ1+α^∫01t1sα^a1+t1Fρ,σλa2−a1−2qα^−t1s+1α^a1+t1Fρ,σλa2−a1−2qα^dt1α^−a1a1+Fρ,σλa2−a12qα^Γ1+α^∫01−ωt1sα^a1+t1Fρ,σλa2−a1−2qα^−t1s+1α^a1+t1Fρ,σλa2−a1−2qα^dt1α^=a1+Fρ,σλa2−a12qα^Γ1+α^Γ1+s+1α^2F¯1α^2q,s+1;s+2;ω−1Fρ,σλa2−a1a1−Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;ω−1Fρ,σλa2−a1a1−1−ωs+1α^Γ1+sα^Γ1+s+1α^×2F¯1α^2q,s+1;s+2;ω−1Fρ,σλa2−a1a1+Λ1α^q,ω,a1,a2;s,(44)Λ4α^q,ω,a1,a2;s≔1Γ1+α^∫1−ω11−t1s+1α^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12qα^dt1α^=a1a1+Fρ,σλa2−a12qα^Γ1+α^∫011−t1s+1α^a1+t1Fρ,σλa2−a1−2qα^dt1α^−a1a1+Fρ,σλa2−a12qα^Γ1+α^∫01−ω1−t1s+1α^a1+t1Fρ,σλa2−a1−2qα^dt1α^=a1+Fρ,σλa2−a12qα^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,1;s+3;ω−1Fρ,σλa2−a1a1−1−ωα^Γ1+α^2F¯1α^2q,1;2;ω−1Fρ,σλa2−a1a1+1−ωs+2α^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;ω−1Fρ,σλa2−a1a1.
A combination of (41)–(44) with (39), we get the desired inequality (37). This gives the proof of Theorem 17.From Theorem17, we get the following corollaries immediately.Corollary 18.
Letω=0. Then, Theorem 17 leads to the following local fractional integral inequality of “Ostrowski type”
(45)∣Qa2−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+α^Γ1+2α^1−1/q×Λ1α^q,0,a1,a2;sQα^a1q+Λ2α^q,0,a1,a2;sQα^a2q1/q,where
(46)Λ1α^q,0,a1,a2;s≔a1+Fρ,σλa2−a12qα^×Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;−Fρ,σλa2−a1a1,Λ2α^q,0,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+α^Γ1+2α^×2F¯1α^2q,2;3;−Fρ,σλa2−a1a1−Λ1α^q,0,a1,a2;s.Corollary 19.
Letω=1. Then, Theorem 17 leads to the following local fractional integral inequality of “Ostrowski type”
(47)∣Qa1−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+α^Γ1+2α^1−1/q×Λ3α^q,1,a1,a2;sQα^a1q+Λ4α^q,1,a1,a2;sQα^a2q1/q,where
(48)Λ3α^q,1,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+1;s+2;−Fρ,σλa2−a1a1−Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;−Fρ,σλa2−a1a1,Λ4α^q,1,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,1;s+3;−Fρ,σλa2−a1a1.Corollary 20.
By means of Corollary18 and Corollary 19. Then, Theorem 17 leads to the following local fractional integral inequality of “trapezoid type”
(49)∣Qa1+Qa22α^−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^2α^a1α^a1+Fρ,σλa2−a1α^Γ1+α^Γ1+2α^1−1/q×Λ1α^q,0,a1,a2;sQα^a1q+Λ2α^q,0,a1,a2;sQα^a2q1/q+Λ3α^q,1,a1,a2;sQα^a1q+Λ4α^q,1,a1,a2;sQα^a2q1/q,where Λ1α^q,0,a1,a2;s,Λ2α^q,0,a1,a2;s,Λ3α^q,1,a1,a2;s, and Λ4α^q,1,a1,a2;s are given in Corollary 18 and Corollary 19, respectively.Corollary 21.
Letω=1/2. Then, Theorem 17 leads to the following local fractional integral inequality of “midpoint type”
(50)∣Q2a1a2a1+a2−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+α^4α^Γ1+2α^1−1/q×Λ1α^q,12,a1,a2;sQα^a1q+Λ2α^q,12,a1,a2;sQα^a2q1/q+Λ3α^q,12,a1,a2;sQα^a1q+Λ4α^q,12,a1,a2;sQα^a2q1/q,where
(51)Λ1α^q,12,a1,a2;s≔12s+2α^a1+Fρ,σλa2−a12qα^×Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;−Fρ,σλa2−a12a1,Λ2α^q,12,a1,a2;s≔122α^a1+Fρ,σλa2−a12qα^Γ1+α^Γ1+2α^×2F¯1α^2q,2;3;Fρ,σλa2−a12a1−Λ1α^q,12,a1,a2;s,Λ3α^q,12,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+sα^Γ1+s+1α^2F¯1α^2q,s+1;s+2;−Fρ,σλa2−a12a1−Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;−Fρ,σλa2−a12a1−12s+1α^Γ1+sα^Γ1+s+1α^×2F¯1α^2q,s+1;s+2;−Fρ,σλa2−a12a1+Λ1α^q,12,a1,a2;s,Λ4α^q,12,a1,a2;s≔a1+Fρ,σλa2−a12qα^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,1;s+3;−Fρ,σλa2−a12a1−12α^Γ1+α^2F¯1α^2q,1;2;−Fρ,σλa2−a12a1+1−ωs+2α^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;−Fρ,σλa2−a12a1.Theorem 22.
Assume that the hypothesis of Lemma16 are satisfied. If Qα^q is a generalized harmonically ψ-s-convex function on IFforp,q>1,p−1+q−1=1. Then, the following inequality for local fractional integrals holds
(52)∣ΩQa1,a1+Fρ,σλa2−a1;ω,α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+sα^Γ1+s+1α^1/q×Λ5α^p,ω,a1,a2;s1/p1−ωs+1α^Qα^a1q+1α^−ωs+1α^Qα^a2q1/q+Λ6α^p,ω,a1,a2;s1/p1α^−1−ωs+1α^Qα^a1q+ωs+1α^Qα^a2q1/q,where
(53)Λ5α^p,ω,a1,a2;s≔a1+Fρ,σλa2−a12pα^1−ωp+1α^Γ1+pα^Γ1+p+1α^×2F¯1α^2p,p+1;p+2;ω−1Fρ,σλa2−a1a1,Λ6α^p,ω,a1,a2;s≔a1+Fρ,σλa2−a12pα^Γ1+p+2α^Γ1+p+3α^2F¯1α^2p,1;p+2;−Fρ,σλa2−a1a1−Γ1+α^2F¯1α^2p,1;2;−Fρ,σλa2−a1a1+1−ωpα^Γ1+p+1α^Γ1+p+2α^×2F¯1α^2p,p+1;p+2;−Fρ,σλa2−a1a1.Proof.
By means of Lemma16 and utilizing the generalized Hölder inequality, we have
(54)∣ΩQa1,a1+Fρ,σλa2−a1;ω,α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^1Γ1+α^∫01−ωt1pα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12pα^dt1α^1/p×1Γ1+α^∫1−ω1Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1qdt1α^1/q+1Γ1+α^∫1−ω11−t1pα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12pα^dt1α^1/p×1Γ1+α^∫1−ω1Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1qdt1α^1/q.
SinceQα^q is generalized harmonically ψ-s-convex function, then we have
(55)1Γ1+α^∫01−ω∣Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1∣dt1α^≤1Γ1+α^∫01−ωt1sα^Qα^a1q+1−t1sα^Qα^a2qdt1α^=Γ1+sα^Γ1+s+1α^1−ωs+1α^Qα^a1q+1α^−ωs+1α^Qα^a2q.
Analogously, we have(56)1Γ1+α^∫1−ω1∣Qα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a1∣dt1α^≤Γ1+sα^Γ1+s+1α^1α^−1−ωs+1α^Qα^a1q+ωs+1α^Qα^a2q.
Settingt1=1−ωη, with the aid of Lemma 7 and simple computations yield
(57)Λ5α^p,ω,a1,a2;s≔1Γ1+α^∫01−ωt1pα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12pα^dt1α^=a1a1+Fρ,σλa2−a12pα^1Γ1+α^∫01−ωt1pα^a1+t1Fρ,σλa2−a1−2pα^dt1α^=a1+Fρ,σλa2−a12pα^1−ωp+1α^1Γ1+α^∫01ηpα^1−ω−1Fρ,σλa2−a1a1−2pα^dt1α^=a1+Fρ,σλa2−a12pα^1−ωp+1α^Γ1+pα^Γ1+p+1α^×2F¯1α^2p,p+1;p+2;ω−1Fρ,σλa2−a1a1.
In a similar way, we have(58)Λ6α^p,ω,a1,a2;s≔1Γ1+α^∫1−ω11−t1pα^a1a1+Fρ,σλa2−a1a1+t1Fρ,σλa2−a12pα^dt1α^=a1+Fρ,σλa2−a12pα^Γ1+p+2α^Γ1+p+3α^2F¯1α^2p,1;p+2;−Fρ,σλa2−a1a1−Γ1+α^2F¯1α^2p,1;2;−Fρ,σλa2−a1a1+1−ωpα^Γ1+p+1α^Γ1+p+2α^×2F¯1α^2p,p+1;p+2;−Fρ,σλa2−a1a1.
A combination of (55)–(58) with (54), we get the desired inequality (52). Hence, the proof is completed.From Theorem22, we get the following corollaries immediately.Corollary 23.
Letω=0. Then, Theorem 17 leads to the following local fractional integral inequality of “Ostrowski type”
(59)∣Qa2−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+sα^Γ1+s+1α^1/q×Λ5α^p,0,a1,a2;s1/pQα^a1q+Qα^a2q1/q,where
(60)Λ5α^p,0,a1,a2;s≔a1+Fρ,σλa2−a12pα^Γ1+rα^Γ1+r+1α^2F¯1α^2p,p+1;p+2;−Fρ,σλa2−a1a1.Corollary 24.
Letω=1. Then, Theorem 17 leads to the following local fractional integral inequality of “Ostrowski type”
(61)Qa1−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+sα^Γ1+s+1α^1/q×Λ6α^p,1,a1,a2;s1/pQα^a1q+Qα^a2q1/q,where
(62)Λ6α^p,1,a1,a2;s≔a1+Fρ,σλa2−a12pα^Γ1+p+2α^Γ1+p+3α^2F¯1α^2p,1;p+2;−Fρ,σλa2−a1a1−Γ1+α^2F¯1α^2p,1;2;−Fρ,σλa2−a1a1.Corollary 25.
By means of Corollary23 and Corollary 24. Then, Theorem 22 leads to the following local fractional integral inequality of “trapezoid type”
(63)∣Qa1+Qa22α^−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^2α^a1α^a1+Fρ,σλa2−a1α^Γ1+sα^Γ1+s+1α^1/q×Λ5α^p,0,a1,a2;s+Λ6α^p,1,a1,a2;sQα^a1q+Qα^a2q1/q.Corollary 26.
Letω=1/2. Then, Theorem 17 leads to the following local fractional integral inequality of “midpoint type”
(64)∣Q2a1a2a1+a2−Fρ,σλa2−a1a1a1+Fρ,σλa2−a1α^Γ1+α^a1Ia1+Fρ,σλa2−a1α^Qzz2α^∣≤Fρ,σλa2−a1α^a1α^a1+Fρ,σλa2−a1α^Γ1+sα^Γ1+s+1α^1/q×Λ5α^p,12,a1,a2;s1/p12s+1α^Qα^a1q+2s+1α^−1α^2s+1α^Qα^a2q1/q+Λ6α^p,12,a1,a2;s1/p2s+1α^−1α^2s+1α^Qα^a1q+12s+1α^Qα^a2q1/q,where
(65)Λ5α^p,12,a1,a2;s≔a1+Fρ,σλa2−a12pα^12p+1α^Γ1+rα^Γ1+r+1α^×2F¯1α^2p,p+1;p+2;−Fρ,σλa2−a12a1,Λ6α^p,12,a1,a2;s≔a1+Fρ,σλa2−a12pα^Γ1+p+2α^Γ1+p+3α^2F¯1α^2p,1;p+2;−Fρ,σλa2−a1a1−Γ1+α^2F¯1α^2p,1;2;−Fρ,σλa2−a1a1+12pα^Γ1+p+1α^Γ1+p+2α^×2F¯1α^2p,p+1;p+2;−Fρ,σλa2−a1a1.


## 4. Applications to Special Means

In this section, we recall the followingα^-type special means for two positive real numbers a1α^ and a2α^ where a1<a2:
(1)
The arithmetic mean(66)Aα^a1,a2=a1α^+a2α^2α^.(2)
The geometric mean(67)Gα^a1,a2=a1α^a2α^.(3)
The harmonic mean(68)ℍα^a1,a2=2a1a2α^a1α^+a2α^.(4)
Ther-logarithmic mean
(69)Lrα^a1,a2=Γ1+rα^Γ1+r+1α^a2r+1α^−a1r+1α^a2−a1α^1r,r∈ℝ\−1,0.These means have huge applications in pure and applied analysis and different type of numerical approximations. However, the following simple relationships are known in the literature.(70)ℍα^a1,a2≤Gα^a1,a2≤Aα^a1,a2.Assume the mappingW:0,∞↦ℝα^,Wz=Γ1+uα^/Γ1+u+1α^zs+1α^,z>0,s≥1 and r≥1. Then, Wα^zr=zsrα^ is generalized harmonically ψ-convex on 0,∞. Therefore, we can obtain the following results for Qz=Γ1+uα^/Γ1+u+1α^xu+1α^.Proposition 27.
For0<a1<a2,u,r>1 and ω∈0,1 with α^∈0,1, we have the following inequality
(71)∣Aα^a1u+1,a2u+1−Γ1+α^Gα^2a1,a1+Fρ,σλa2−a1Lu−1α^u−1a1,a1+Fρ,σλa2−a1∣≤Γ1+u+1α^Γ1+uα^Fρ,σλa2−a1α^Gα^22,a1,a1+Fρ,σλa2−a1Γ1+α^Γ1+2α^1−1/q×Λ1α^q,0,a1,a2;sa1quα^+Λ2α^q,0,a1,a2;sa2quα^1/q+Λ3α^q,1,a1,a2;sa1quα^+Λ4α^q,1,a1,a2;sa2quα^1/q,where Λ1α^q,0,a1,a2;s,Λ2α^q,0,a1,a2;s,Λ3α^q,1,a1,a2;s, and Λ4α^q,1,a1,a2;s are given in Corollary 18 and Corollary 19, respectively.Proof.
TakingQz=Γ1+uα^/Γ1+u+1α^zs+1α^,u≥1 for z>0 in Corollary 20, then we get the immediate consequence.Proposition 28.
For0<a1<a2,s,q1>1 and ω∈0,1 with α^∈0,1, we have the following inequality
(72)∣ℍα^s+1a1,a1+Fρ,σλa2−a1−Γ1+α^Gα^2a1,a1+Fρ,σλa2−a1Ls−1α^s−1a1,a1+Fρ,σλa2−a1∣≤Γ1+u+1α^Γ1+uα^Fρ,σλa2−a1α^Gα^2a1a1+Fρ,σλa2−a1Γ1+sα^Γ1+s+1α^1/q×Λ5α^p,12,a1,a2;s1/p12s+1α^a1quα^+2s+1α^−1α^2s+1α^a2quα^1/q+Λ6α^p,12,a1,a2;s1/p2s+1α^−1α^2s+1α^a1quα^+12s+1α^a2quα^1/q,where Λ5α^p,1/2,a1,a2;s and Λ6α^p,1/2,a1,a2;s are defined in Corollary 26.Proof.
TakingQz=Γ1+uα^/Γ1+u+1α^zu+1α^,u≥1 for z>0 in Corollary 26, then we get the immediate consequence.


## 5. Applications to Mittag-Leffler Function

Takingλ=1,1,⋯, with ρ=δ1,Rδ1>0 and σ=1, then from Theorems 13 to 22, the following results hold.Theorem 29.
LetQ:ΩF=a1,a1+Eδ1a2−a1⊂ℝ\0↦ℝα^0≤α^<1 be a generalized harmonically ψ-s-convex function where a1,a1+Eδ1a2−a1∈ΩF,Eδ1a2−a1>0 such that Q∈Dα^a1,a1+Eδ1a2−a1 and Qα^∈ℂα^a1,a1+Eδ1a2−a1. Then, the following inequalities hold:
(73)2s−1α^Γ1+α^Q2a1a1+Eδ1a2−a12a1+Eδ1a2−a1≤a1α^a1+Eδ1a2−a1α^Eδ1a2−a1α^a1Ia1+Eδ1a2−a1α^Qzz2α^≤Γ1+sα^Γ1+s+1α^Qa1+Qa2.Theorem 30.
Forρ,σ>0,λ=λmm=0∞ a bounded sequence of real numbers and Q:IF∘=a1,a1+Eδ1a2−a1⊂ℝ\0↦ℝα^0<α^≤1 (IF∘ is the interior of IF), where a1,a1+Eδ1a2−a1∈IF∘,Eδ1a2−a1>0 such that Q∈Dα^a1,a1+Eδ1a2−a1 and Qα^∈ℂα^a1,a1+Eδ1a2−a1.IfQα^q1 is a generalized harmonically ψ-s-convex function on IF for p,q>1,p−1+q−1=1. Then, the following inequality for local fractional integrals holds
(74)ΩQa1,a1+Eδ1a2−a1;ω,α^≤Eδ1a2−a1α^a1α^a1+Eδ1a2−a1α^Γ1+α^Γ1+2α^1−1/q×1−ω21−1qα^Λ1α^q,ω,a1,a2;sQα^a1q+Λ2α^q,ω,a1,a2;sQα^a2q1/q+ω21−1qα^Λ3α^q,ω,a1,a2;sQα^a1q+Λ4α^q,ω,a1,a2;sQα^a2q1/q,where
(75)Λ1α^q,ω,a1,a2;s≔1−ωs+2α^a1+Eδ1a2−a12qα^×Γ1+s+1α^Γ1+s+2α^2F¯1α^2q,s+2;s+3;ω−1Eδ1a2−a1a1,Λ2α^q,ω,a1,a2;s≔1−ω2α^a1+Eδ1a2−a12qα^Γ1+α^Γ1+2α^×2F¯1α^2q,2;3;ω−1Eδ1a2−a1a1−Λ1α^q,ω,a1,a2;s,Λ3α^q,ω,a1,a2;s≔a1+Eδ1a2−a12qα^Γ1+sα^Γ1+s+1α^2F¯1α^2q,s+1;s+2;ω−1Eδ1a2−a1a1−Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;ω−1Eδ1a2−a1a1−1−ωs+1α^Γ1+sα^Γ1+s+1α^×2F¯1α^2q,s+1;s+2;ω−1Eδ1a2−a1a1+Λ1α^q,ω,a1,a2;s,Λ4α^q,ω,a1,a2;s≔a1+Eδ1a2−a12qα^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,1;s+3;ω−1Eδ1a2−a1a1−1−ωα^Γ1+α^2F¯1α^2q,1;2;ω−1Eδ1a2−a1a1+1−ωs+2α^Γ1+s+2α^Γ1+s+3α^2F¯1α^2q,s+2;s+3;ω−1Eδ1a2−a1a1.


## 6. Conclusion

On the development of this work, we have introduced the notion of generalized harmonicallyψ-convex functions in fractal domain. Adopting this novel approach, we derived a new identity that has been correlated with some novel and well-known midpoint type, Ostrowski type, and trapezoid type inequalities in the relative literature have been introduced. For generalized harmonically ψ-s-convex functions, we investigated bounds for Hermite-Hadamard type inequalities for local fractional differentiation in the frame of Raina’s function, which direct to several noted estimates computed by [40]. In addition to this, the findings and the techniques in this work are entirely new. This study demonstrates methods that can be utilized to examine the much to explore in the area of local fractional calculus, geometric aided design by proposing the variation in the fractal parameters. Hence, the proposed methodology is a powerful tool for solving local fractional linear equations of physical importance.



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