# Generalized Fractional Hadamard and Fejér–Hadamard Inequalities for Generalized Harmonically Convex Functions

Authors: Chahn Yong Jung; Muhammad Yussouf; Yu-Ming Chu; Ghulam Farid; Shin Min Kang
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we define a new function, namely, harmonicallyα,h−m-convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically α,h−m-convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically h−m-convex, harmonically α,m-convex, and related functions and for already known fractional integral operators.


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## Body

## 1. Introduction and Preliminary Results

Fractional integral inequalities are the generalizations and extensions of classical integral inequalities using fractional integral/derivative operators. There are many well-known inequalities which have been extended for fractional calculus operators: for example, Hadamard, Minkowski, Ostrowski, Grüss, Ostrowski–Grüss, and Chebyshev inequalities have been extensively studied in recent decades, see [1–8]. The aim of this paper is to present the Hadamard and the Fejér–Hadamard inequalities for fractional integral operators for harmonically α,h−m-convex functions. We begin from the fractional integral operators defined by Andrić et al. in [2] containing an extended generalized Mittag-Leffler function in their kernels.Definition 1.
(see [2]). Let ω,τ,δ,ρ,c∈ℂ, ℜτ,ℜδ>0, and ℜc>ℜρ>0 with p≥0, σ,r>0, and 0<k≤r+σ. Let f∈L1a,b and x∈a,b. Then, the generalized fractional integral operators εσ,τ,δ,ω,a+ρ,r,k,cf and εσ,τ,δ,ω,b−ρ,r,k,cf are defined as follows:(1)εσ,τ,δ,ω,a+ρ,r,k,cfx;p=∫axx−tτ−1Eσ,τ,δρ,r,k,cωx−tσ;pftdt,εσ,τ,δ,ω,b−ρ,r,k,cfx;p=∫xbt−xτ−1Eσ,τ,δρ,r,k,cωt−xσ;pftdt,where(2)Eσ,τ,δρ,r,k,ct;p=∑n=0∞βpρ+nk,c−ρcnktnβρ,c−ρΓσn+τδnr,is the extended generalized Mittag-Leffler function.
In [9], Farid defined elegantly the unified integral operators as follows:Definition 2.
Letf,g:a,b⟶ℝ, 0<a<b, be the functions such that f is positive and f∈L1a,b and g is differentiable and strictly increasing. Also, let ϕ/x be an increasing function on a,∞ and ω,τ,δ,ρ,c∈ℂ, ℜτ,ℜδ>0, and ℜc>ℜρ>0 with p≥0, σ,r>0, and 0<k≤r+σ. Then, for x∈a,b, the integral operators Fσ,τ,δ,ω,a+ϕ,ρ,r,k,cgf and Fσ,τ,δ,ω,b−ϕ,ρ,r,k,cgf are defined as follows:(3)Fσ,τ,δ,ω,a+ϕ,ρ,r,k,cgfx;p=∫axϕgx−gtgx−gtEσ,τ,δρ,r,k,cωgx−gtσ;pftdgt,Fσ,τ,δ,ω,b−ϕ,ρ,r,k,cgfx;p=∫xbϕgt−gxgt−gxEσ,τ,δρ,r,k,cωgt−gxσ;pftdgt.
The following definition of extended generalized fractional integral operators containing an extended Mittag-Leffler function in the kernel can be extracted by takingϕx=xτ in Definition 2.Definition 3.
Letf,g:a,b⟶ℝ, 0<a<b, be the functions such that f is positive and f∈L1a,b and g is differentiable and strictly increasing. Also, let ω,τ,δ,ρ,c∈ℂ, ℜτ,ℜδ>0, and ℜc>ℜρ>0 with p≥0, σ,r>0, and 0<k≤r+σ. Then, for x∈a,b, the generalized fractional integral operators are defined as follows:(4)ϒσ,τ,δ,ω,a+ρ,r,k,cgfx;p=∫axgx−gtτ−1Eσ,τ,δρ,r,k,cωgx−gtσ;pftdgt,(5)ϒσ,τ,δ,ω,b−ρ,r,k,cgfx;p=∫xbgt−gxτ−1Eσ,τ,δρ,r,k,cωgt−gxσ;pftdgt.
Fractional integral operators (4) and (5) produce particularly various well-known fractional integral operators.
The Hadamard inequality gives the bounds of integral mean of a convex function on a closed interval. It is equivalently defined by the convex function. Convex functions have a significant role in the field of mathematical inequalities.Definition 4.
A functionf:a,b⟶ℝ is said to be convex if(6)ftx+1−ty≤tfx+1−tfy,holds for all x,y∈a,b and t∈0,1.
The Hadamard inequality is stated in the following theorem.Theorem 1.
Letf:a,b⟶ℝ be a convex function. Then, the following inequality holds:(7)fa+b2≤1b−a∫abfxdx≤fa+fb2.The Fejér–Hadamard inequality proved by Fejér in [10] generalizes the Hadamard inequality, and it is given as follows:Theorem 2.
Letf:a,b⟶ℝ be a convex function and g:a,b⟶ℝ be nonnegative, integrable, and symmetric about a+b/2. Then, the following inequality holds:(8)fa+b2∫abgxdx≤∫abfxgxdx≤fa+fb2∫abgxdx.The definition of the convex function motivates the researchers to define new notions and their consequences. A notion of the harmonically convex function is defined in [11].Definition 5.
LetI be an interval such that I⊆ℝ+; a function f:I⟶ℝ is said to be harmonically convex if(9)fabta+1−tb≤tfb+1−tfa,holds for all a,b∈I and t∈0,1.
This notation is further extended to the harmonicallyα,m-convex function in [12] and harmonically h-convex function in [13].Definition 6.
(see [12]). A function f:0,b⟶ℝ, b>0, is said to be harmonically α,m-convex if f is nonnegative and(10)fmxymty+1−tx≤tαfx+m1−tαfy,holds for all x,y∈0,b, t, α∈0,1, and m∈0,1.Definition 7.
(see [13]). Let h:0,1⊆J⟶ℝ be a nonnegative function. A function f:J⊆ℝ+⟶ℝ is said to be harmonically h-convex if(11)fxyty+1−tx≤htfx+h1−tfy,holds for all x,y∈J and t∈0,1.
In the upcoming section, we define a new generalized notion, namely, harmonicallyα,h−m-convex function, which unifies all known definitions of harmonically convex functions. By using this new definition, we will prove Hadamard and Fejér–Hadamard inequalities for generalized fractional integral operators involving an extended generalized Mittag-Leffler function. Also, results for harmonically h−m-convex functions and harmonically α,m-convex functions are deduced.


## 2. Main Results

First, we define the harmonicallyα,h−m-convex function as follows:Definition 8.
Leth:0,1⊆J⟶ℝ be a nonnegative function. A function f:J⊆ℝ+⟶ℝ is said to be harmonically α,h−m-convex if(12)fmxymty+1−tx≤htαfx+mh1−tαfy,holds for all x,y∈J, t, α∈0,1, and m∈0,1.
It unifies the definitions of harmonicallyα,m-convexity and harmonically h-convexity of functions. For different specific choices of α,h,andm, almost all kinds of well-known harmonically convex functions can be obtained, see the remark as follows.Remark 1.
(i)
Ifht=t, then a harmonically α,m-convex function can be obtained [12](ii)
Ifα=1 and ht=ts, then a harmonically s,m-convex function can be obtained [14](iii)
Ifα=1 and ht=t, then a harmonically m-convex function can be obtained [14](iv)
Ifα=ht=m=1, then a harmonically P-function can be obtained [13](v)
Ifα=1, ht=ts, and m=1, then a harmonically s-convex function can be obtained [13](vi)
Ifα=1, ht=1/t, and m=1, then a harmonically Godunova–Levin function can be obtained [13](vii)
Ifα=1, ht=1/ts, and m=1, then a harmonically s-Godunova–Levin function can be obtained [13](viii)
If we setm=1 and α=1, then a harmonically h-convex function can be achieved [13](ix)
By puttingα=1, ht=t, and m=1, then a harmonically convex function can be obtained [11]
We give another new notion of the harmonicallyh−m-convex function by setting α=1 in Definition 8 as follows.Definition 9.
Leth:0,1⊆J⟶ℝ be a nonnegative function. A function f:J⊆ℝ+⟶ℝ is said to be harmonically h−m-convex if(13)fmxymty+1−tx≤htfx+mh1−tfy.
Now, we apply Definition8 and operators (4) and (5) to find the Hadamard and Fejér–Hadamard inequalities.
In the whole paper, we have used the following notations frequently:(14)ℱb,τa+ω,f=ϒσ,τ,δ,ω,a+ρ,r,k,cgfb;p,ℱa,τb−ω,f=ϒσ,τ,δ,ω,b−ρ,r,k,cgfa;p.Theorem 3.
Letf,g:a,b⟶ℝ, 0<a<b, range g⊂a,b, be the functions such that f is positive and f∈L1a,b and g is differentiable and strictly increasing. If f is harmonically α,h−m-convex on a,b, then for operators (4) and (5), we have(15)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,1≤h12αℱg−11/m2gb,τg−11/mga−mσω′,f∘ψ+m1−2τh1−12αℱg−11/ga,τg−11/mgb+ω′,f∘ψ≤mgb−gam2gagbτ,mh12αfgam2+mh1−12αfgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;phtαdt+h12αfgb+mh1−12αfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−tαdt,where ω′=ωmgab/mgb−gaσ and ψt=1/gt for all t∈1/b,1/a.Proof.
Sincef is harmonically α,h−m-convex, for all x,y∈a,b, the following inequality holds:(16)f2mgxgymgy+gx≤h12αfgx+mh1−12αfgy.
By settinggx=mgagb/tgb+1−tga/m and gy=mgagb/tga+m1−tgb in (16), we have(17)f2m2gagbga+mgb≤h12αfmgagbtgb+1−tga/m+mh1−12αfmgagbtga+m1−tgb.
Multiplying (17) by tτ−1Eσ,τ,δρ,r,k,cωtσ;p and integrating over 0,1, we get(18)f2m2gagbga+mgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pdt≤h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbtgb+1−tga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbtga+m1−tgbdt.
By settinggx=tgb+1−tga/m/mgagb and gy=tga+m1−tgb/mgagb in (18) and using (4) and (5), the first inequality of (15) can be obtained. On the contrary, from the harmonically α,h−m-convexity of f, we have(19)h12αfmgagbtgb+1−tga/m+mh1−12αfmgagbtga+m1−tgb≤h12αh1−tαfgb+mhtαfgam2+mh1−12αh1−tαfga+mhtαfgb=mh12αfgam2+mh1−12αfgbhtα+h12αfgb+mh1−12αfgah1−tα.
Multiplying (19) by tτ−1Eσ,τ,δρ,r,k,cωtσ;p and then integrating over 0,1, we get(20)h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbtgb+1−tga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbtga+m1−tgbdt≤mh12αfgam2+mh1−12αfgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;phtαdt+h12αfgb+mh1−12αfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;phtαh1−tαdt.
By settinggx=tgb+1−tga/m/mgagb and gy=tga+m1−tgb/mgagb in (20) and using (4) and (5), the second inequality of (15) can be obtained.Remark 2.
(i)
By settingp=0, α=m=1, ht=t, and g=I, Theorem 3.1 of [15] is obtained(ii)
By settingα=m=1, ht=t, and g=I, Theorem 2.1 of [16] is obtained(iii)
By settingω=p=0, α=m=1, ht=t, and g=I, Theorem 4 of [6] is obtainedTheorem 4.
Letf,g,ℏ:a,b⟶ℝ, 0<a<b, range g, range ℏ⊂a,b, be the functions such that f is positive and f∈L1a,b, g is differentiable and strictly increasing, and ℏ is nonnegative and integrable. If f is harmonically α,h−m-convex and f1/gx=f1/1/ga+1/mgb−mgx, then for operators (4) and (5), we have(21)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψ≤h12α+mh1−12αℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψf∘ψ≤mgb−gam2gagbτmh12αfgam2+mh1−12αfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mhtαdt+h12αfgb+mh1−12αfga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mh1−tαdt,where ω′=ωmgab/mgb−gaσ and ψt=1/gt for all t∈1/b,1/a.Proof.
Multiplying (17) by tτ−1ℏmgagb/tgb+1−tga/mEσ,τ,δρ,r,k,cωtσ;p and then integrating over 0,1, we get(22)f2m2gagbga+mgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mdt≤h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mfmgagbtgb+1−tga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mfmgagbtga+m1−tgbdt.
By choosinggx=tgb+1−tga/m/mgagb, that is, mgagb/tga+m1−tgb=1/1/ga1/mgb−mgx, in (22) and using (4) and (5) and the condition f1/gx=f1/1/ga1/mgb−mgx, the first inequality of (21) can be obtained. For the second inequality of (21), multiplying (19) by tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagb/tgb+1−tga/m and then integrating over 0,1, we get(23)h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mfmgagbtgb+1−tga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mfmgagbtga+m1−tgbdt≤mh12αfgam2+mh1−12αfgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;p×ℏmgagbtgb+1−tga/mhtαdt+h12αfgb+mh1−12αfga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;phtαℏmgagbtgb+1−tga/mh1−tαdt.
By settinggx=tgb+1−tga/m/mgagb in (23) and using (4) and (5) and the condition f1/gx=f1/1/ga1/mgb−mgx, the second inequality of (21) can be obtained. □Remark 3.
(i)
By settingp=0, α=m=1, ht=tℏx=1, and g=I, Theorem 3.1 of [15] is obtained(ii)
By settingg=I, α=m=1, ht=t, and ℏx=1, Theorem 2.1 of [16] is obtained(iii)
By settingω=p=0, ℏx=1, α=m=1, ht=t, and g=I, Theorem 4 of [6] is obtained(iv)
By settingω=p=0, τ=1, α=m=1, ht=t, and g=I, Theorem 8 of [4] is obtained(v)
By settingω=p=0, τ=1, ℏx=1, α=m=1, ht=t, and g=I, Theorem 4 of [17] is obtainedTheorem 5.
Letf,g:a,b⟶ℝ, 0<a<b, range g⊂a,b, be the functions such that f is positive and f∈L1a,b and g is differentiable and strictly increasing. If f is harmonically α,h−m-convex on a,b, then for operators (4) and (5), we have(24)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1≤h12αℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,f∘ψ+m1−2τh1−12αℱg−11/ga,τg−1ga+mgb/2mgagb+2σω′,f∘ψ≤mgb−ga2m2gagbτmh12αfgam2+mh1−12αfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pht2αdt+h12αfgb+mh1−12αfga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−t2αdt,ω′=ωmgabmgb−gaσ,where ψt=1/gt and t∈1/b,1/a.Proof.
By settinggx=mgagb/t/2gb+1−t/2ga/m and gy=mgagb/t/2ga+m1−t/2gb in (16), we have(25)f2m2gagbga+mgb≤h12αfmgagbt/2gb+1−t/2ga/m+mh1−12αfmgagbt/2ga+m1−t/2gb.
Multiplying (25) by tτ−1Eσ,τ,δρ,r,k,cωtσ;p and then integrating over 0,1, we get(26)f2m2gagbga+mgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pdt≤h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbt/2gb+1−t/2ga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbt/2ga+m1−t/2gbdt.
By settinggx=t/2gb+1−t/2ga/m/mgagb and gy=t/2ga+m1−t/2gb/mgagb in (26) and using (4) and (5), the first inequality of (24) can be obtained. On the contrary, from the harmonically α,h−m-convexity of f, we have(27)h12αfmgagbt/2gb+1−t/2ga/m+mh1−12αfmgagbt/2ga+m1−t/2gb≤h12αh1−t2αfgb+mht2αfgam2+mh1−12αh1−t2αfga+mht2αfgb=mh12αfgam2+mh1−12αfgbht2α+h12αfgb+mh1−12αfgah1−t2α.
Multiplying (27) by tτ−1Eσ,τ,δρ,r,k,cωtσ;p and integrating over 0,1, we get(28)h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbt/2gb+1−t/2ga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pfmgagbt/2ga+m1−t/2gbdt≤mh12αfgam2+mh1−12αfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pht2αdt+h12αfgb+mh1−12αfga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−t2αdt.
By settinggx=t/2gb+1−t/2ga/m/mgagb and gy=t/2ga+m1−t/2gb/mgagb in (28) and using (4) and (5), the second inequality of (24) can be obtained.Remark 4.
(i)
By settingp=0, α=m=1, ht=t, and g=I, Theorem 3.3 of [15] is obtained(ii)
By settingα=m=1, ht=t, and g=I, Theorem 2.3 of [16] is obtainedTheorem 6.
Letf,g,ℏ:a,b⟶ℝ, 0<a<b, range g, range ℏ⊂a,b, be the functions such that f is positive and f∈L1a,b, g is differentiable and strictly increasing, and ℏ is nonnegative and integrable. If f is harmonically α,h−m convex and f1/gx=f1/1/ga1/mgb−mgx, then for operators (4) and (5), we have(29)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψ≤h12α+mh1−12αℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψf∘ψ≤mgb−ga2m2gagbτmh12αfgam2+mh1−12αfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mht2αdt+h12αfgb+mh1−12αfga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mh1−t2αdt,where ω′=ωmgab/mgb−gaσ and ψt=1/gt for all t∈1/b,1/a.Proof.
Multiplying (25) by tτ−1ℏmgagb/t/2gb+1−t/2ga/mEσ,τ,δρ,r,k,cωtσ;p and integrating over 0,1, we get(30)f2m2gagbga+mgb∫01tτ−1ℏmgagbt/2gb+1−t/2ga/mEσ,τ,δρ,r,k,cωtσ;pdt≤h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mfmgagbt/2gb+1−t/2ga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mfmgagbt/2ga+m1−t/2gbdt.
By choosinggx=mgagb/t/2gb+1−t/2ga/m in (30) and using (4) and (5) and the condition f1/gx=f1/1/ga+1/mgb−mgx, the first inequality of (29) can be obtained. On the contrary, by the harmonically α,h−m-convexity of f, multiplying (27) by tτ−1ℏmgagb/t/2gb+1−t/2ga/mEσ,τ,δρ,r,k,cωtσ;p and integrating over 0,1, we get(31)h12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mfmgagbt/2gb+1−t/2ga/mdt+mh1−12α∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/m×fmgagbt/2ga+m1−t/2gbdt≤mh12αfgam2+mh1−12αfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mht2αdt+h12αfgb+mh1−12αfga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mh1−t2αdt.
By choosinggx=mgagb/t/2gb+1−t/2ga/m in (31) and using (4) and (5) and the condition f1/gx=f1/1/ga+1/mgb−mgx, the second inequality of (29) can be obtained.Remark 5.
(i)
By settingp=0, α=m=1, ht=t, and g=I, Theorem 3.6 of [15] is obtained(ii)
By settingα=m=1, ht=t, and g=I, Theorem 5 of [16] is obtained


### 2.1. Results for Harmonicallyh−m-Convex Functions

By settingα=1 in Theorems 3–6, the results for harmonically h−m-convex functions are obtained as follows.Theorem 7.
Under the assumptions of Theorem3, the following inequality holds for harmonically h−m-convex functions:(32)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,1≤h12ℱg−11/m2gb,τg−11/mga−mσω′,f∘ψ+m1−2τℱg−11/ga,τg−11/mgb+ω′,f∘ψ≤h12mgb−gam2gagbτmfgam2+mfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;phtdt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−tdt.Theorem 8.
Under the assumptions of Theorem4, the following inequality holds for harmonically h−m-convex functions:(33)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψ≤h121+mℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψf∘ψ≤h12mgb−gam2gagbτmfam2+mfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mhtdt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mh1−tdt.Theorem 9.
Under the assumptions of Theorem5, the following inequality holds for harmonically h−m-convex functions:(34)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1≤h12ℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,f∘ψ+m1−2τℱg−11/ga,τg−1ga+mgb/2mgagb+2σω′,f∘ψ≤h12mgb−ga2m2gagbτmfgam2+mfgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pht2dt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−t2dt.Theorem 10.
Under the assumptions of Theorem6, the following inequality holds for harmonically h−m-convex functions:(35)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψ≤h121+mℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψf∘ψ≤h12mgb−ga2m2gagbτmfgam2+mfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb1−t/2+ga/mht2dt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mh1−t2dt.


### 2.2. Results for Harmonicallyα,m-Convex Functions

By settinght=t in Theorems 3–6, the results for harmonically α−m-convex functions are obtained as follows.Theorem 11.
Under the assumptions of Theorem3, the following inequality holds for harmonically α,m-convex functions:(36)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,1≤12αℱg−11/m2gb,τg−11/mga−mσω′,f∘ψ+m1−2τ2α−1ℱg−11/ga,τg−11/mgb+ω′,f∘ψ≤12αmgb−gam2gagbτmfgam2+m2α−1fgb×ℱg−11/m2gb,τg−11/mga−mσω′,1+fgb+m2α−1fga×ℱg−11/m2gb,τg−11/mga−mσω′,1−ℱg−11/m2gb,τ+αg−11/mga−mσω′,1.Theorem 12.
Under the assumptions of Theorem4, the following inequality holds for harmonically α,m-convex functions:(37)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψ≤12α1+mℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψf∘ψ≤12αmgb−gam2gagbτmfgam2+m2α−1fgb×∫01tτ+α−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mdt+fgb+m2α−1fga×∫01tτ−1−tτ+α−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mdt.Theorem 13.
Under the assumptions of Theorem5, the following inequality holds for harmonically α,m-convex functions:(38)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1≤12αℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,f∘ψ+m1−2τℱg−11/ga,τg−1ga+mgb/2mgagb+2σω′,f∘ψ≤12αmgb−ga2m2gagbτmfgam2+m2α−1fgb×12αℱg−11/m2gb,τ+αg−1ga+mgb/2m2gagb−2mσω′,1+fgb+m2α−1fga×ℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1−12αℱg−11/m2gb,τ+αg−1ga+mgb/2m2gagb−2mσω′,1.Theorem 14.
Under the assumptions of Theorem6, the following inequality holds for harmonically α,m-convex functions:(39)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψ≤12α1+mℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψf∘ψ≤12αmgb−ga2m2gagbτmfgam2+m2α−1fgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mt2αdt+fgb+m2α−1fga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/m1−t2αdt.


## 2.1. Results for Harmonicallyh−m-Convex Functions

By settingα=1 in Theorems 3–6, the results for harmonically h−m-convex functions are obtained as follows.Theorem 7.
Under the assumptions of Theorem3, the following inequality holds for harmonically h−m-convex functions:(32)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,1≤h12ℱg−11/m2gb,τg−11/mga−mσω′,f∘ψ+m1−2τℱg−11/ga,τg−11/mgb+ω′,f∘ψ≤h12mgb−gam2gagbτmfgam2+mfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;phtdt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−tdt.Theorem 8.
Under the assumptions of Theorem4, the following inequality holds for harmonically h−m-convex functions:(33)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψ≤h121+mℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψf∘ψ≤h12mgb−gam2gagbτmfam2+mfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mhtdt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mh1−tdt.Theorem 9.
Under the assumptions of Theorem5, the following inequality holds for harmonically h−m-convex functions:(34)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1≤h12ℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,f∘ψ+m1−2τℱg−11/ga,τg−1ga+mgb/2mgagb+2σω′,f∘ψ≤h12mgb−ga2m2gagbτmfgam2+mfgb∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pht2dt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;ph1−t2dt.Theorem 10.
Under the assumptions of Theorem6, the following inequality holds for harmonically h−m-convex functions:(35)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψ≤h121+mℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψf∘ψ≤h12mgb−ga2m2gagbτmfgam2+mfgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb1−t/2+ga/mht2dt+fgb+mfga∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mh1−t2dt.


## 2.2. Results for Harmonicallyα,m-Convex Functions

By settinght=t in Theorems 3–6, the results for harmonically α−m-convex functions are obtained as follows.Theorem 11.
Under the assumptions of Theorem3, the following inequality holds for harmonically α,m-convex functions:(36)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,1≤12αℱg−11/m2gb,τg−11/mga−mσω′,f∘ψ+m1−2τ2α−1ℱg−11/ga,τg−11/mgb+ω′,f∘ψ≤12αmgb−gam2gagbτmfgam2+m2α−1fgb×ℱg−11/m2gb,τg−11/mga−mσω′,1+fgb+m2α−1fga×ℱg−11/m2gb,τg−11/mga−mσω′,1−ℱg−11/m2gb,τ+αg−11/mga−mσω′,1.Theorem 12.
Under the assumptions of Theorem4, the following inequality holds for harmonically α,m-convex functions:(37)f2m2gagbga+mgbℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψ≤12α1+mℱg−11/m2gb,τg−11/mga−mσω′,ℏ∘ψf∘ψ≤12αmgb−gam2gagbτmfgam2+m2α−1fgb×∫01tτ+α−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mdt+fgb+m2α−1fga×∫01tτ−1−tτ+α−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbtgb+1−tga/mdt.Theorem 13.
Under the assumptions of Theorem5, the following inequality holds for harmonically α,m-convex functions:(38)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1≤12αℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,f∘ψ+m1−2τℱg−11/ga,τg−1ga+mgb/2mgagb+2σω′,f∘ψ≤12αmgb−ga2m2gagbτmfgam2+m2α−1fgb×12αℱg−11/m2gb,τ+αg−1ga+mgb/2m2gagb−2mσω′,1+fgb+m2α−1fga×ℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,1−12αℱg−11/m2gb,τ+αg−1ga+mgb/2m2gagb−2mσω′,1.Theorem 14.
Under the assumptions of Theorem6, the following inequality holds for harmonically α,m-convex functions:(39)f2m2gagbga+mgbℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψ≤12α1+mℱg−11/m2gb,τg−1ga+mgb/2m2gagb−2mσω′,ℏ∘ψf∘ψ≤12αmgb−ga2m2gagbτmfgam2+m2α−1fgb×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/mt2αdt+fgb+m2α−1fga×∫01tτ−1Eσ,τ,δρ,r,k,cωtσ;pℏmgagbt/2gb+1−t/2ga/m1−t2αdt.


## 3. Concluding Remarks

The presented work includes two generalized fractional versions of the Hadamard inequality (stated in Theorems3 and5) as well as two generalized fractional versions of the Fejér–Hadamard inequality (stated in Theorems 4 and6) for a new generalized harmonic convexity. Their connections with already published results are shown in Remarks 2, 3, 4, and 5. Also, we work out these inequalities for harmonically α,m-convex and h−m-convex functions. The reader can obtain a series of Hadamard and Fejér–Hadamard inequalities for fractional integral operators containing Mittag-Leffler functions for several kinds of convex functions.



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