# Trapezium-Type Inequalities fork-Fractional Integral via New Exponential-Type Convexity and Their Applications

Authors: Artion Kashuri; Sajid Iqbal; Saad Ihsan Butt; Jamshed Nasir; Kottakkaran Sooppy Nisar; Thabet Abdeljawad
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, the authors investigated the concept ofs,m-exponential-type convex functions and their algebraic properties. New generalizations of Hermite–Hadamard-type inequality for the s,m-exponential-type convex function ψ and for the products of two s,m-exponential-type convex functions ψ and ϕ are proved. Many refinements of the (H–H) inequality via s,m-exponential-type convex are obtained. Finally, several new bounds for special means and new error estimates for the trapezoidal and midpoint formula are provided as well. The ideas and techniques of this paper may stimulate further research in different areas of pure and applied sciences.


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## Body

## 1. Introduction

Theory of convexity also played a significant role in the development of theory of inequalities. Many famously known results in inequalities theory can be obtained using the convexity property of the functions. Hermite–Hadamard’s double inequality is one of the most intensively studied results involving convex functions. This result provides us necessary and sufficient condition for a function to be convex. It is also known as classical equation of H–H inequality.The Hermite–Hadamard inequality assert that if a functionψ:J⊂ℜ⟶ℜ is convex in J for a1,a2∈J and a1<a2, then(1)ψa1+a22≤1a2−a1∫a1a2ψχdχ≤ψa1+ψa22.Interested readers can refer to [1–29].Definition 1 (see [30]).
A functionψ:0,+∞⟶ℜ is said to be s-convex in the second sense for a real number s∈0,1 or ψ belongs to the class of Ks2, if(2)ψχa1+1−χa2≤χsψa1+1−χsψa2,holds for all a1,a2∈0,+∞ and χ∈0,1.
As-convex function was introduced in Breckner’s article in [30], and a number of properties and connections with s-convexity in the first sense are discussed in [10]. Usually, convexity means for s-convexity when s=1. Dragomir et al. proved a variant of Hadamard’s inequality in [5], which holds for s-convex functions in the second sense. In the last decade, many mathematicians added the rich literature in the field of mathematical inequalities involving fractional calculus (see [8, 9, 13, 15, 22, 25, 28, 31]).
Toader introduced the class ofm-convex functions in [26].Definition 2 (see [26]).
A functionψ:0,a2⟶ℜ, a2>0, is said to be m-convex, where m∈0,1, if(3)ψχθ1+m1−χθ2≤χψθ1+m1−χψθ2holds ∀θ1,θ2∈0,a2 and χ∈0,1. Otherwise, ψ is m-concave if −ψ is m-convex.
In a recent paper, Eftekhari [6] defined the class of s,m-convex functions in the second sense as follows.Definition 3.
A functionψ:0,+∞⟶ℜ is said to be s,m-convex for some fixed real numbers s,m∈0,1 if(4)ψχa1+m1−χa2≤χsψa1+m1−χsψa2holds ∀a1,a2∈0,+∞ and χ∈0,1.
Fractional integral inequalities are useful to find the uniqueness of solutions for certain fractional partial differential equations (see [19, 32]).
Letψ∈La1,a2. Then, Riemann–Liouville fractional integrals of order α>0 with a1≥0 are defined as follows:(5)Ja1+αψx=1Γα∫a1xx−χα−1ψχdχ,x>a1,Ja2−αψx=1Γα∫xa2χ−xα−1ψχdχ,x<a2.
For further details, one may see [14, 16, 18].
In [7, 17], there is given definition of k−fractional Riemann–Liouville integrals.
Letψ∈La1,a2. Then, k−fractional integrals of order α,k>0 with a1≥0 are defined as follows:(6)Ja1+αkψx=1kΓkα∫a1xx−χα/k−1ψχdχ,x>a1,Ja2−αkψx=1kΓkα∫xa2χ−xα/k−1ψχdχ,x<a2,where Γkα is the k−gamma function defined as(7)Γkα=∫0+∞χα−1e−χk/kdχ.
We can notice that(8)Γkα+k=αΓkα,Ja1+01ψx=Ja2−01ψx=ψx.
By choosingk=1, the above k−fractional integrals yield Riemann–Liouville integrals.
Also that, the incomplete gamma functionγϑ,θ is defined for ϑ>0 and θ≥0 by integral(9)γϑ,θ=∫0θe−μμϑ−1dμ.
The gamma functionΓϑ is defined for ϑ>0 by integral(10)Γϑ=∫0+∞e−μμϑ−1dμ.
Motivated by above results and literatures, we will give first in Section2 the concept of s,m-exponential-type convex function, and we will study some of their algebraic properties. In Section 3, we will prove new generalizations of Hermite–Hadamard-type inequality for the s,m-exponential-type convex function ψ and for the products of two s,m-exponential-type convex functions ψ and ϕ. In Section 4, we will obtain some refinements of the (H–H) inequality for functions whose first derivative in absolute value at certain power is s,m-exponential-type convex. In Section 5, some new bounds for special means and error estimates for the trapezoidal and midpoint formula will be provided. In Section 6, a briefly conclusion will be given as well.


## 2. Some Algebraic Properties ofs,m-Exponential-Type Convex Functions

In this section, we will introduce a new definition, calleds,m-exponential-type convex function, and we will study some basic algebraic properties of it.Definition 4.
A nonnegative functionψ:J⟶ℜ is said s,m-exponential-type convex for some fixed s,m∈0,1 if(11)ψχθ1+m1−χθ2≤esχ−1ψθ1+me1−χs−1ψθ2holds for all θ1,θ2∈J and χ∈0,1.Remark 1.
Form=s=1, we get exponential- type convexity given by İşcan in [11].Remark 2.
s,m-exponential-type convex functions for some fixed m∈0,1 and s∈ln2.5,1 have the range 0,+∞.Proof.
Letθ∈J be arbitrary for some fixed m∈0,1 and s∈ln2.5,1. Using the Definition 4 for χ=1, we have(12)ψθ≤es−1ψθ⟹es−2ψθ≥0⟹ψθ≥0.Lemma 1.
For allχ∈0,1 and for some fixed m∈0,1 and s∈ln2.5,1, the following inequalities esχ−1≥χs and e1−χs−1≥1−χs hold.Proof.
The proof is evident.Proposition 1.
Every nonnegatives,m-convex function is s,m-exponential-type convex function for some fixed m∈0,1 and s∈ln2.5,1.Proof.
By using Lemma1, for some fixed m∈0,1 and s∈ln2.5,1, we have(13)ψχθ1+m1−χθ2≤χsψθ1+m1−χsψθ2≤esχ−1ψθ1+me1−χs−1ψθ2.Theorem 1.
Letψ,ϕ:a1,a2⟶ℜ. If ψ and ϕ are s,m-exponential-type convex functions for some fixed s,m∈0,1, then the following holds:(1)
ψ+ϕ is s,m-exponential-type convex function.(2)
For nonnegative real numberc, cψ is s,m-exponential-type convex function.Proof.
By Definition4, for some fixed s,m∈0,1, the proof is obvious.Theorem 2.
Letψ:0,a2⟶J be m-convex function for a2>0 and some fixed m∈0,1 and ϕ:J⟶ℜ is nondecreasing and s,m-exponential-type convex function for some fixed s,m∈0,1. Then, for the same fixed numbers s,m∈0,1, the function ϕ°ψ:0,a2⟶ℜ is s,m-exponential-type convex.Proof.
For allθ1,θ2∈0,a2 and χ∈0,1 and for some fixed numbers s,m∈0,1, we have(14)ϕ°ψχθ1+m1−χθ2=ϕψχθ1+m1−χθ2≤ϕχψθ1+m1−χψθ2≤esχ−1ϕ°ψθ1+me1−χs−1ϕ°ψθ2.Theorem 3.
Letψi:a1,a2⟶ℜ be an arbitrary family of s,m-exponential-type convex functions for the same fixed s,m∈0,1 and let ψθ=supiψiθ. If A=θ∈a1,a2:ψθ<+∞≠∅, then A is an interval and ψ is s,m-exponential-type convex function on A.Proof.
For allθ1,θ2∈A and χ∈0,1 and for the same fixed numbers s,m∈0,1, we have(15)ψχθ1+m1−ψθ2=supiψiχθ1+m1−χθ2≤supiesχ−1ψiθ1+me1−χs−1ψiθ2≤esχ−1supiψiθ1+me1−χs−1supiψiθ2=esχ−1ψθ1+me1−χs−1ψθ2<+∞.
This means simultaneously thatA is an interval, and ψ is s,m-exponential-type convex function on A.Theorem 4.
If the functionψ:a1,a2⟶ℜ is s,m-exponential-type convex for some fixed s,m∈0,1, then ψ is bounded on a1,ma2.Proof.
LetL=maxψa1,ψa2/m and x∈a1,a2 be an arbitrary point for some fixed m∈0,1. Then, there exists χ∈0,1 such that x=χa1+1−χa2. Thus, since esχ≤es and e1−χs≤es for some fixed s∈0,1, we have(16)ψx=ψχa1+1−χa2≤esχ−1ψa1+me1−χs−1ψa2m≤es−1L+mes−1L=m+1Les−1=M.
We have shown thatψ is bounded above from real number M. Interested reader can also prove the fact that ψ is bounded below using the same idea as in Theorem 4 in [11].


## 3. New Generalizations of (H–H)-Type Inequality

In this section, we will establish some new generalizations of Hermite–Hadamard-type inequality for thes,m-exponential-type convex function ψ and for the products of two s,m-exponential-type convex functions ψ and ϕ.Theorem 5.
Letψ:a1,ma2⟶ℜ be s,m-exponential-type convex function for some fixed s,m∈0,1 and a1<ma2. If ψ∈L1a1,ma2, then(17)1es/2−1ψa1+ma22≤1ma2−a1∫a1ma2ψxdx+m∫a1/ma2ψxdx≤es−s−1sψa1+ψa2+mψa1m2+ψa2.Proof.
Let denote, respectively,(18)θ1=χa1+m1−χa2,θ2=1−χa1m+χa2,∀χ∈0,1.
Usings,m-exponential-type convexity of ψ, we have(19)ψa1+ma22=ψθ1+mθ22=ψχa1+m1−χa2+1−χa1+mχa22≤es/2−1ψχa1+m1−χa2+ψ1−χa1+mχa2.
Now, integrating on both sides in the last inequality with respect toχ over 0,1, we get(20)ψa1+ma22≤es/2−1×∫01ψχa1+m1−χa2dχ+∫01ψ1−χa1m+χa2dχ=es/2−1ma2−a1∫a1ma2ψxdx+m∫a1/ma2ψxdx,which completes the left-side inequality. For the right-side inequality, using s,m-exponential-type convexity of ψ, we obtain(21)1ma2−a1∫a1ma2ψxdx+m∫a1ma2ψxdx=∫01ψχa1+m1−χa2dχ+∫01ψ1−χa1m+χa2dχ≤∫01esχ−1ψa1+me1−χs−1ψa2dχ+∫01esχ−1ψa2+me1−χs−1ψa1m2dχ=es−s−1sψa1+ψa2+mψa1m2+ψa2,which gives the right-side inequality.Corollary 1.
By takingm=s=1 in Theorem 5, we get (Theorem 5, [11]).Theorem 6.
Assume thatψ,ϕ:a1,ma2⟶ℜ are, respectively, s1,m and s2,m-exponential-type convex functions for the same fixed m∈0,1 and for some fixed s1,s2∈0,1, where s1<s2 and a1<ma2. If ψ,ϕ are synchronous functions and ψ,ϕ,ψϕ∈L1a1,ma2, then(22)1ma2−a1∫a1ma2ψθdθ⋅1ma2−a1∫a1ma2ϕθdθ≤1ma2−a1∫a1ma2ψθϕθdθ≤As1,s2Mma1,a2+Bs1,s2Na1,a2,where(23)Mma1,a2=ψa1ϕa1+m2ψa2ϕa2,Na1,a2=ψa1ϕa2+ψa2ϕa1,As1,s2=es1+s2+s1+s2−1s1+s2−s1es2−1+s2es1−1s1s2,Bs1,s2=es2−es1s2−s1−s1es2−1+s2es1−1s1s2+1.Proof.
Let us denoteθ=χa1+m1−χa2 for all χ∈0,1. Using the property of the s1,m and s2,m-exponential-type convex functions ψ and ϕ, respectively, we have(24)ψχa1+m1−χa2≤es1χ−1ψa1+me1−χs1−1ψa2,ϕχa1+m1−χa2≤es2χ−1ga1+me1−χs2−1ϕa2.
Multiplying above inequalities on both sides, we get(25)ψχa1+m1−χa2ϕχa1+m1−χa2≤es1χ−1ψa1+me1−χs1−1ψa2×es2χ−1ga1+me1−χs2−1ϕa2=es1χ−1es2χ−1ψa1ϕa1+mes1χ−1e1−χs2−1ψa1ϕa2+es2χ−1e1−χs1−1ψa2ϕa1+m2e1−χs1−1e1−χs2−1ψa2ϕa2.
Applying Chebyshev integral inequality (see [23]), we obtain(26)∫01ψχa1+m1−χa2ϕχa1+m1−χa2dχ≥∫01ψχa1+m1−χa2dχ⋅∫01ϕχa1+m1−χa2dχ.
Changing the variable of integration, we get(27)1ma2−a1∫a1ma2ψθϕθdθ≥1ma2−a1∫a1ma2ψθdθ⋅1ma2−a1∫a1ma2ϕθdθ,which completes the left-side inequality. For the right-side inequality, integrating on both sides of the inequality (25) with respect to χ over 0,1, we have(28)1ma2−a1∫a1ma2ψθdθ⋅1ma2−a1∫a1ma2ϕθdθ≤∫01es1χ−1es2χ−1dχψa1ϕa1+m∫01es1χ−1e1−χs2−1dχψa1ϕa2+∫01es2χ−1e1−χs1−1dχψa2ϕa1+m2∫01e1−χs1−1e1−χs2−1dχψa2ϕa2=As1,s2Mma1,a2+Bs1,s2Na1,a2,which give the right-side inequality.


## 4. Refinements of (H–H)-Type Inequality fork−Fractional Integral

In this section, we will obtain some refinements of the (H–H) inequality vias,m-exponential-type convex functions.Lemma 2.
Suppose0<w≤1 and a mapping ψ:wa1,a2⟶ℜ is differentiable on wa1,a2 with 0<a1<a2. If ψ′∈L1wa1,a2, then the following equality for k−fractional integral holds:(29)ψwa1+α/kψa2α/k+1−Γkα+ka2−wa1α/kkJa2−αψwa1=a2−wa1α/k+1∫01αk+1χα/k−1ψ′w1−χa1+χa2dχ,whereα,k>0 and Γ⋅ is the Euler Gamma function.Proof.
Using the integrating by parts, we have(30)a2−wa1α/k+1∫01αk+1χα/k−1ψ′w1−χa1+χa2dχ=a2−wa1α/k+1∫01αk+1χα/kψ′w1−χa1+χa2dχ−∫01ψ′w1−χa1+χa2dχ=a2−wa1α/k+1αk+1χα/kψw1−χa1+χa2a2−wa1|01−∫01ψw1−χa1+χa2a2−wa1αkχα/k−1dχ−ψw1−χa1+χa2a2−wa1|01=a2−wa1α/k+1αk+1×ψa2a2−wa1−αka2−wa1∫01χα/k−1ψw1−χa1+χa2dχ=ψwa1+α/kψa2α/k+1−Γkα+ka2−wa1α/kkJa2−αψwa1,which completes the proof.Lemma 3.
Suppose0<w≤1 and a mapping ψ:wa1,a2⟶ℜ is differentiable on wa1,a2 with 0<a1<a2. If ψ′∈L1wa1,a2, then the following equality for k−fractional integral holds:(31)ψwa1+ψa2w+1−Γkα+kw+1a2−wa1α/kJa1+αkψa2+Ja2−αkψwa1=a2−wa1w+1∫01χα/k−1−χα/kψ′w1−χa1+χa2dχ.Proof.
Using the integrating by parts, we have(32)a2−wa1w+1∫01χα/k−1−χα/kψ′w1−χa1+χa2dχ=a2−wa1w+1∫01χα/kψ′w1−χa1+χa2dχ−∫011−χα/kψ′w1−χa1+χa2dχ=a2−wa1w+1I1−I2,where(33)I1=∫01χα/kψ′w1−χa1+χa2dχ=χα/kψw1−χa1+χa2a2−wa1|01−∫01ψw1−χa1+χa2a2−wa1αkχα/k−1dχ=ψa2a2−wa1−αka2−wa1∫01χα/k−1ψw1−χa1+χa2dχ=ψa2a2−wa1−Γkα+ka2−wa1α/k+1kJa2−αψwa1,(34)I2=∫011−χα/kψ′w1−χa1+χa2dχ=1−χα/kψw1−χa1+χa2a2−wa1|01−∫01ψw1−χa1+χa2a2−wa1αk1−χα/k−1−1dχ=−ψwa1a2−wa1+αka2−wa1∫011−χα/k−1ψw1−χa1+χa2dχ=−ψwa1a2−wa1+Γkα+ka2−wa1α/k+1kJa1+αψa2.
Combining equations (33) and (34) in (32) and multiplying by a2−wa1/w+1, we get (31), which completes the proof.Theorem 7.
Suppose0<w≤1 and a mapping ψ:0,a2/mw⟶ℜ is differentiable on 0,a2/mw with 0<a1<a2. If ψ′q is s,m-exponential-type convex on 0,a2/mw for q>1 and q−1+p−1=1, then for some fixed s,m∈0,1, then the following inequality for k−fractional integral holds:(35)ψwa1+α/kψa2α/k+1−Γkα+ka2−wa1α/kkJa2−αψwa1≤a2−wa1α/k+1U1α,k,p+U2α,k,p1/p×es−s−1sψ′wa1q+mψ′a2mq1/q,where(36)U1α,k,p=∫01/αα/k+1k1−αk+1χα/kpdχ,U2α,k,p=∫1/αα/k+1k1αk+1χα/k−1pdχ.Proof.
From Lemma2, Hölder’s inequality and s,m-exponential-type convexity of ψ′q, we have(37)ψwa1+α/kψa2α/k+1−Γkα+ka2−wa1α/kkJa2−αψwa1≤a2−wa1α/k+1∫01αk+1χα/k−1ψ′w1−χa1+χa2dχ≤a2−wa1α/k+1∫01αk+1χα/k−1pdχ1/p∫01ψ′w1−χa1+χa2qdχ1/q≤a2−wa1α/k+1∫01αk+1χα/k−1pdχ1/p×∫01e1−χs−1ψ′wa1q+mesχ−1ψ′a2mqdχ1/q=a2−wa1α/k+1U1α,k,p+U2α,k,p1/p×es−s−1sψ′wa1q+mψ′a2mq1/q,which completes the proof.Theorem 8.
Suppose0<w≤1 and a mapping ψ:0,a2/mw⟶ℜ is differentiable on 0,a2/mw with 0<a1<a2. If ψ′q is s,m-exponential-type convex on 0,a2/mw for q≥1, then for some fixed s,m∈0,1, then the following inequality for k−fractional integral holds:(38)ψwa1+α/kψa2α/k+1−Γkα+ka2−wa1α/kkJa2−αψwa1≤a2−wa1α/k+12αkα/k+1k/α+11−1/q×ψ′wa1q−2αkα/k+1k/α+1−2ses1−1/αα/k+1k−αk+1essα/kγαk+1,1αα/k+1k+αk+1ess−α/kγ1−1/αα/k+1kαk+1,1αα/k+1k+1s+mψ′a2m2es/αα/k+1ks−2αkα/k+1k/α+1−αk+11−sα/kγαk+1,1αα/k+1k+αk+11−sα/kγ1−1/αα/k+1kαk+1,1αα/k+1k−ess1/q.Proof.
From Lemma2, power mean inequality and s,m-exponential-type convexity of ψ′q, we have(39)ψwa1+α/kψa2α/k+1−Γkα+ka2−wa1α/kkJa2−αψwa1≤a2−wa1α/k+1∫01αk+1χα/k−1ψ′w1−χa1+χa2dχ≤a2−wa1α/k+1∫01αk+1χα/k−1dχ1−1/q×∫01αk+1χα/k−1ψ′w1−χa1+χa2qdχ1/q≤a2−wa1α/k+1∫01αk+1χα/k−1dχ1−1/q×∫01αk+1χα/k−1e1−χs−1ψ′wa1q+mesχ−1ψ′a2mqdχ1/q=a2−wa1α/k+12αkα/k+1k/α+11−1/q×ψ′wa1q−2αkα/k+1k/α+1−2ses1−1/αα/k+1k−αk+1essα/kγαk+1,1αα/k+1k+αk+1essα/kγ1−1αα/k+1kαk+1,1αα/k+1k+1s+mψ′a2m2es/αα/k+1ks−2αkα/k+1k/α+1−αk+11−sα/kγαk+1,1αα/k+1k+αk+11−sα/kγ1−1/αα/k+1kαk+1,1αα/k+1k−ess1/q,which completes the proof.Theorem 9.
Suppose0<w≤1 and a mapping ψ:0,a2/m⟶ℜ is differentiable on 0,a2/m with 0<a1<a2. If ψ′q is s,m-exponential-type convex on 0,a2/m for q>1 and q−1+p−1=1, then for some fixed s,m∈0,1, then the following inequality for k−fractional integral holds:(40)ψwa1+ψa2w+1−Γkα+kw+1a2−wa1α/kJa1+αkψa2+Ja2−αkψwa1≤2a2−wa1w+1kαp+k1/pes−s−1sψ′wa1q+mψ′a2mq1/q.Proof.
From Lemma3, Hölder’s inequality and s,m-exponential-type convexity of ψ′q, we have(41)ψwa1+ψa2w+1−Γkα+kw+1a2−wa1α/kJa1+αkψa2+Ja2−αkψwa1≤a2−wa1w+1∫01χα/k−1−χα/kψ′w1−χa1+χa2dχ≤a2−wa1w+1∫01χα/kψ′w1−χa1+χa2dχ+∫011−χα/kψ′w1−χa1+χa2dχ≤a2−wa1w+1∫01χα/kpdχ1/p∫01ψ′w1−χa1+χa2qdχ1/q+∫011−χα/kpdχ1/p∫01ψ′w1−χa1+χa2qdχ1/q≤a2−wa1w+1∫01χα/kpdχ1/p×∫01e1−χs−1ψ′wa1q+mesχ−1ψ′a2mqdχ1/q+∫011−χα/kpdχ1/p×∫01e1−χs−1ψ′wa1q+mesχ−1ψ′a2mqdχ1/q=2a2−wa1w+1kαp+k1/pes−s−1sψ′wa1q+mψ′a2mq1/q,which completes the proof.Theorem 10.
Suppose0<w≤1 and a mapping ψ:0,a2/m⟶ℜ is differentiable on 0,a2/m with 0<a1<a2. If ψ′q is s,m-exponential-type convex on 0,a2/m for q≥1, then for some fixed s,m∈0,1, then the following inequality for k−fractional integral holds:(42)ψwa1+ψa2w+1−Γkα+kw+1a2−wa1α/kJa1+αkψa2+Ja2−αkψwa1≤a2−wa1w+1kα+k1−1/q×ψ′wa1qΓαk+1−Γαk+1,sess−α/k−1−1α/k+1+mψ′a2mqΓα/k+1,−s−Γα/k+1−sα/ks−1α/k+11/q+ψ′wa1q−1α/k−1s−α/k−1Γαk+1−Γαk+1,−s−1α/k+1+mψ′a2mqα/k+1esks−α/k−1Γα/k+1,s−Γα/k+1k+α−1α/k+11/q.Proof.
From Lemma3, power mean inequality and s,m-exponential-type convexity of ψ′q, we have(43)ψwa1+ψa2w+1−Γkα+kw+1a2−wa1α/kJa1+αkψa2+Ja2−αkψwa1≤a2−wa1w+1∫01χα/k−1−χα/kψ′w1−χa1+χa2dχ≤a2−wa1w+1×∫01χα/kψ′w1−χa1+χa2dχ+∫011−χα/kψ′w1−χa1+χa2dχ≤a2−wa1w+1∫01χα/kdχ1−1/q∫01χα/kψ′w1−χa1+χa2qdχ1/q+∫011−χα/kdχ1−1/q∫011−χα/kψ′w1−χa1+χa2qdχ1/q≤a2−wa1w+1∫01χα/kdχ1−1/q×∫01χα/ke1−χs−1ψ′wa1q+mesχ−1ψ′a2mqdχ1/q+∫011−χα/ke1−χs−1ψ′wa1q+mesχ−1ψ′a2mqdχ1/q=a2−wa1w+1kα+k1−1/q×ψ′wa1qΓαk+1−Γαk+1,sess−α/k−1−1α/k+1+mψ′a2mqΓα/k+1,−s−Γα/k+1−sα/ks−1α/k+11/q+ψ′wa1q−1α/k−1s−α/k−1Γαk+1−Γαk+1,−s−1α/k+1+mψ′a2mqα/k+1esks−α/k−1Γα/k+1,s−Γα/k+1k+α−1α/k+11/q,which completes the proof.


## 5. Applications

Let consider the following two special means for different positive-real numbersa1<a2.(1)
The arithmetic mean:(44)Aa1,a2=a1+a22.(2)
The generalizedlog-mean:(45)ℒla1,a2=a2l+1−a1l+1l+1a2−a11/l;l∈ℜ\−1,0.Dragomir et al. [5] have proved that, for s∈0,1, where 1≤l≤1/s, the function fx=xls,x>0 is s-convex function. Then, from Proposition 1, it is also s-exponential convex function for some fixed s∈ln2.5,1.Using Section4, we have the following interesting results:Proposition 2.
Let0<a1<a2, 0<w≤1, and q>1 such that p−1+q−1=1. Then, for some fixed s∈ln2.5,1, where 1≤l≤1/s, we have(46)Awa1ls,a2ls−ℒlslswa1,a2≤lsa2−wa1p2×1p+11/pes−s−1s1/qA1/qwa1ls−1q,a2ls−1q.Proof.
Consider thes-exponential convex function ψx=xls,x>0, and using Theorem 7 for α=1=k, we obtain the required result.Proposition 3.
Let0<a1<a2, 0<w≤1 and q≥1. Then, for some fixed s∈ln2.5,1, where 1≤l≤1/s, we get(47)Awa1ls,a2ls−ℒlslswa1,a2≤lsa2−wa141−1/q×2s−2es+8es/2−s2−2s−42s21/qA1/qwa1ls−1q,a2ls−1q.Proof.
Consider thes-exponential convex function ψx=xls,x>0, and using Theorem 8 for α=1=k, the result is obvious.Proposition 4.
Let0<a1<a2, 0<w≤1 and q>1, such that p−1+q−1=1. Then, for some fixed s∈ln2.5,1, where 1≤l≤1/s, we obtain(48)2w+1Awa1ls,a2ls−ℒlslswa1,a2≤q2lsa2−wa1w+1×1p+11/pes−s−1s1/qA1/qwa1ls−1q,a2ls−1q.Proof.
Consider thes-exponential convex function ψx=xls,x>0, and using Theorem 9 for α=1=k, the result is evident.Proposition 5.
Let0<a1<a2, 0<w≤1 and q≥1. Then, for some fixed s∈ln2.5,1, where 1≤l≤1/s, we have(49)2w+1Awa1ls,a2ls−ℒlslswa1,a2≤lsa2−wa12w+1×2s−2es+8es/2−s2−2s−42s21/qA1/qwa1ls−1q,a2ls−1q.Proof.
Consider thes-exponential convex function ψx=xls,x>0, and using Theorem 10 for α=1=k, we obtain the required result.
At the end, let consider some applications of the integral inequalities obtained above, to find new bounds for the trapezoidal and midpoint formula.
Fora2>0, let U:0=χ0<χ1<,⋯,<χn−1<χn=a2 is a partition of 0,a2.
We denote(50)TU,ψ=∑i=0n−1ψχi+ψχi+12hi,∫0a2ψxdx=TU,ψ+ℛU,ψ,where ℛU,ψ is the remainder term and hi=χi+1−χi for i=0,1,2,…,n−1.
Using above notations, we are in position to prove the following error estimations.Proposition 6.
Suppose a mappingψ:0,a2⟶ℜ is differentiable on 0,a2 with a2>0. If ψ′q is s-exponential-type convex on 0,a2 for q>1 and q−1+p−1=1, then for some fixed s∈0,1, the remainder term satisfies the following error estimation:(51)ℛU,ψ≤121p+11/pes−s−1s1/q×∑i=0n−1hi2ψ′χiq+ψ′χi+1q1/q.Proof.
Using Theorem7 on subinterval χi,χi+1 of closed interval 0,a2, for all i=0,1,2,…,n−1 and w=α=k=m=1, we get(52)ψχi+ψχi+12hi−∫χiχi+1ψxdx≤121p+11/pes−s−1s1/q×hi2ψ′χiq+ψ′χi+1q1/q.Summing inequality (52) over i from 0 to n−1 and using the property of modulus, we obtain the desired inequality (51).Proposition 7.
Suppose a mappingψ:0,a2⟶ℜ is differentiable on 0,a2 with a2>0. If ψ′q is s-exponential-type convex on 0,a2 for q≥1, then for some fixed s∈0,1, the remainder term satisfies the following error estimation:(53)ℛU,ψ≤es−s−12s1/q∑i=0n−1hi2ψ′χiq+ψ′χi+1q1/q.Proof.
Apply the same technique as in Proposition6 but using Theorem 8.


## 6. Conclusion

In this article, the authors showed new generalizations of trapezium-type inequality for the new class of functions, the so-calleds,m-exponential-type convex function ψ and for the products of two s,m-exponential-type convex functions ψ and ϕ. We have obtained refinements of the (H–H) inequality for functions using s,m-exponential-type convex and founded new bounds for special means and for the error estimates for the trapezoidal and midpoint formula. We hope that current work will attract the attention of researchers working in mathematical analysis, fractional calculus, quantum calculus, postquantum calculus, and other related fields.



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