Some new fractional integral inequalities for generalized relative semi-$\mathbf{m}$-(r; h₁, h₂)-preinvex mappings via generalized Mittag-Leffler function

Abstract

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

Citation

Kashuri, A. and Liko, R. (2020), "Some new fractional integral inequalities for generalized relative semi-$\mathbf{m}$-(r; h₁, h₂)-preinvex mappings via generalized Mittag-Leffler function", Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 41-55. https://doi.org/10.1016/j.ajmsc.2018.12.003

1. Introduction

The following double inequality is known as Hermite–Hadamard inequality.

Theorem 1.1.

Let f:I⊆ℝ→ℝ be a convex mapping on an interval I of real numbers and a,b∈I with a<b. Then the subsequent double inequality holds:

(1.1)f(a+b2)≤1b−a∫abf(x)dx≤f(a)+f(b)2.

For recent results concerning Hermite–Hadamard type inequalities through various classes of convex functions the readers are referred to [3–5,7,8,12–,20,21,23–25,29,32] and the references mentioned in these papers.

Let us recall some special functions and evoke some basic definitions as follows.

Definition 1.2

([20]). Let f∈L[a,b]. The Riemann–Liouville integrals Ja+αf and Jb−αf of order α>0 with a≥0 are defined by

Ja+αf(x)=1Γ(α)∫ax(x−t)α−1f(t)dt, x>a

and

Jb−αf(x)=1Γ(α)∫xb(t−x)α−1f(t)dt, b>x,

where Γ(α)=∫0+∞e−uuα−1du. Here Ja+0f(x)=Jb−0f(x)=f(x).

Note that α=1, the fractional integral reduces to the classical integral.

Definition 1.3

([27]). Let μ,ν,k,l,γ be positive real numbers and ω∈ℝ. Then the generalized fractional integral operators containing Mittag-Leffler function ϵμ,ν,l,ω,a+γ,δ,k and ϵμ,ν,l,ω,b−γ,δ,k for a real valued continuous function f are defined by:

(1.2)(ϵμ,ν,l,ω,a+γ,δ,kf)(x)=∫ax(x−t)v−1Eμ,ν,lγ,δ,k(ω(x−t)μ)f(t)dt

and

(ϵμ,ν,l,ω,b−γ,δ,kf)(x)=∫xb(t−x)v−1Eμ,ν,lγ,δ,k(ω(t−x)μ)f(t)dt,

where the function Eμ,ν,lγ,δ,k is the generalized Mittag-Leffler function defined as

(1.3)Eμ,ν,lγ,δ,k(t)=∑0∞(γ)kntnΓ(μn+ν)(δ)ln

and (a)n is the Pochhammer symbol, it defined as

(a)n=a(a+1)(a+2)⋅…⋅(a+n−1), (a)0=1.

For ω=0 in (1.2), integral operator ϵμ,ν,l,ω,a+γ,δ,k reduces to the Riemann–Liouville fractional integral operator.

In [27,30] properties of generalized integral operator and generalized Mittag-Leffler functions are studied in detail. In [27] it is proved that Eμ,ν,lγ,δ,k(t) is absolutely convergent for k<l+μ. Let S be the sum of series of absolute terms of Eμ,ν,lγ,δ,k(t). We will use this property of Mittag-Leffler function in sequel.

Definition 1.4

([1]). A set K⊆ℝn is said to be invex with respect to the mapping Λ:K×K→ℝn, if x+tΛ(y,x)∈K for every x,y∈K and t∈[0,1].

Definition 1.5

([7]). A non-negative function f:I⊆ ℝ → [0,+∞) is said to be P-function, if

f(tx+(1−t)y)≤ f(x)+f(y), ∀x,y∈I, t∈[0,1].

Definition 1.6

([22]). Let h :[0,1]→ℝ be a non-negative function and h ≠ 0. The function f on the invex set K is said to be h-preinvex with respect to Λ, if

(1.4)f(x+tΛ(y,x)) ≤ h(1−t) f(x)+h(t) f(y)

for each x,y∈ K and t∈[0,1] where f(⋅)>0.

Definition 1.7

([31]). Let f :K⊆ ℝ→ℝ be a non-negative function. A function f :K→ℝ is said to be a tgs-convex on K if the inequality

(1.5)f((1−t)x+ty)≤t(1−t)[f(x)+f(y)]

holds for all x,y ∈ K and t ∈(0,1).

Definition 1.8

([19]). A function f :I⊆ ℝ→ ℝ is said to be MT-convex, if it is non-negative and ∀x,y∈ I and t∈(0,1) satisfies the subsequent inequality:

(1.6)f(tx+(1−t)y)≤t21−tf(x)+1−t2tf(y).

Definition 1.9

([25]). A function: f :I⊆ ℝ → ℝ is said to be m-MT-convex, if f is positive and for ∀x,y∈I, and t∈(0,1), among m∈(0,1], satisfies the following inequality

(1.7)f(tx+m(1−t)y)≤t21−tf(x)+m1−t2tf(y).

Definition 1.10

([8]). A set K⊆ℝn is named as m-invex with respect to the mapping Λ:K×K→ℝn for some fixed m ∈ (0,1], if mx+tΛ(y,mx)∈ K holds for each x,y ∈ K and any t ∈ [0,1].

Remark 1.11.

In Definition 1.10, under certain conditions, the mapping Λ(y,mx) could be reduced to Λ(y,x). For example when m=1, then the m-invex set degenerates an invex set on K.

Definition 1.12

([26]). Let K⊆ℝ be an open m-invex set with respect to the mapping Λ : K × K→ℝ and h1,h2:[0,1]→[0,+∞). A function f:K→ℝ is said to be generalized (m,h1,h2)-preinvex, if

(1.8)f(mx+tΛ(y,mx))≤mh1(t)f(x)+h2(t)f(y)

is valid for all x,y ∈ K and t ∈ [0,1], for some fixed m ∈ (0,1].

Motivated by the above literatures, the main objective of this paper is to establish in Section 2, some new fractional integral inequalities for generalized relative semi-m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function. It is pointed out that some new special cases will be deduced from main results of the paper. Also we will see that these inequalities have some connections with known integral inequalities. In Section 3, some applications to special means for different positive real numbers will be given.

2. Main results

The following definitions will be used in this section.

Definition 2.1.

Let m:[0,1]→(0,1] be a function. A set K ⊆ ℝn is named as m-invex with respect to the mapping Λ : K × K → ℝn, if m(t)x+ξΛ(y,m(t)x)∈ K holds for each x,y ∈ K and any t,ξ ∈ [0,1].

Remark 2.2.

In Definition 2.1, under certain conditions, the mapping Λ(y,m(t)x) for any t,ξ ∈ [0,1] could be reduced to Λ(y,mx). For example when m(t)=m for all t∈[0,1], then the m-invex set degenerates to an m-invex set on K.

We next introduce the notion of generalized relative semi-m-(r;h1,h2)-preinvex mappings.

Definition 2.3.

Let K ⊆ ℝ be an open m-invex set with respect to the mapping Λ : K×K→ℝ. Suppose h1,h2:[0,1]→[0,+∞),ψ:I→K are continuous functions and m:[0,1]→(0,1]. A mapping f :K→(0,+∞) is said to be generalized relative semi-m-(r;h1,h2)-preinvex, if

(2.1)f(m(t)ψ(x)+ξΛ(ψ(y),m(t)ψ(x))) ≤ [m(ξ)h1(ξ)fr(x)+h2(ξ)fr(y)]1r

holds for all x,y ∈ I and t,ξ ∈ [0,1], where r ≠ 0.

Remark 2.4.

In Definition 2.3, if we choose m=m=r=1, this definition reduces to the definition considered by Noor in [23] and Preda et al. in [11].

Remark 2.5.

In Definition 2.3, if we choose m=m=r=1 and ψ(x)=x, then we get Definition 1.12.

Remark 2.6.

Let us discuss some special cases in Definition 2.3 as follows.

1. Taking h1(t)=h2(t)=1, then we get the generalized relative semi-(m,P)-preinvex mappings.

2. Taking h1(t)=(1−t)s and h2(t)=ts for s∈(0,1], then we get the generalized relative semi-(m,s)-Breckner-preinvex mappings.

3. Taking h1(t)=(1−t)−s and h2(t)=t−s for s∈(0,1], then we get the generalized relative semi-(m,s)-Godunova–Levin–Dragomir-preinvex mappings.

4. Taking h1(t)=h(1−t) and h2(t)=h(t), then we get the generalized relative semi-(m,h)-preinvex mappings.

5. Taking h1(t)=h2(t)=t(1−t), then we get the generalized relative semi-(m,tgs)-preinvex mappings.

6. Taking h1(t)=1−t2t and h2(t)=t21−t , then we get the generalized relative semi-m-MT-preinvex mappings.

It is worth mentioning here that to the best of our knowledge all the special cases discussed above are new in the literature.

For establishing our main results we need to prove the following lemma.

Lemma 2.7.

Let ψ : I → K and g :K → ℝ are continuous functions and m :[0,1]→(0,1]. Suppose K=[m(t)ψ(a),m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))] ⊆ ℝ be an open m-invex subset with respect to Λ : K×K→ℝ for Λ(ψ(b),m(t)ψ(a))>0 and ∀t ∈ [0,1]. Assume that f : K→ ℝ be a differentiable mapping on K∘. If f′,g ∈ L(K), then the following equality for ν > 0 holds:

(2.2)(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν×[f(m(t)ψ(a))+f(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a)))]−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν−1×g(ξ)Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν−1×g(ξ)Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ=∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds)νf'(ξ)dξ−∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν×f'(ξ)dξ.

We denote

(2.3)If,g,E,Λ,ψ,m(ν,a,b):=∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν×f'(ξ)dξ−∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν×f′(ξ)dξ.

Proof. Integrating by parts, we get

If,g,E,Λ,ψ,m(ν,a,b)=(∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds)νf(ξ)|m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds)v−1× g(ξ)Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ−(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)νf(ξ)|m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)v−1×g(ξ)Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ=(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)ν×[f(m(t)ψ(a))+f(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a)))]−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds)v−1×g(ξ)Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds)v−1×g(ξ)Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ.

This completes the proof of the lemma.

Using Lemma 2.7, we now state the following theorems for the corresponding version for power of first derivative.

Theorem 2.8.

Let h1,h2 :[0,1]→[0,+∞),ψ:I→K and g:K→ℝ are continuous functions and m:[0,1]→(0,1]. Suppose K=[m(t)ψ(a),m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))]⊆ℝ be an open m-invex subset with respect to Λ :K×K→ℝ for Λ(ψ(b),m(t)ψ(a))>0 and ∀t∈[0,1]. Assume that f : K→(0,+∞) be a differentiable mapping on K∘ such that f′,g∈ L(K). If (f′(x))q is generalized relative semi-m-(r;h1,h2)-preinvex mapping, 0<r≤1,k<l+μ,q>1,p−1+q−1=1 and ||g||∞=sups∈K|g(s)|, then the following inequality for ν>0 holds:

(2.4)|If,g,E,Λ,ψ,m(ν,a,b)|≤2||g||∞νSνΛν+1(ψ(b),m(t)ψ(a))pν+1p×(f′(a))rqI1r(h1(ξ);m(ξ),r)+(f′(b))rqI2r(h2(ξ);r)rq,

where

I1(h1(ξ);m(ξ),r):=∫01m1r(ξ)h11r(ξ)dξ, I2(h2(ξ);r):=∫01h21r(ξ)dξ.

Proof. From Lemma 2.7, the generalized relative semi-m-(r;h1,h2)-preinvexity of (f′(x))q, Hölder inequality, Minkowski inequality, absolute convergence of Mittag-Leffler function, properties of the modulus, the fact g(s) ≤ ‖g‖∞,∀s ∈ K and changing the variable u=m(t)ψ(a)+ξΛ(ψ(b),m(t)ψ(a)),∀t∈[0,1], we have

|If,g,E,Λ,ψ,m(ν,a,b)|≤∫m(t)ψ(a)(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds|v×|f′(ξ)|dξ+∫m(t)ψ(a)(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds|v×|f′(ξ)|dξ≤(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫m(t)ψ(a)ξg(s)Eμ,ν,lγ,δ,k(ωsμ)ds|pvdξ)1p×(∫m(t)ψ(a)(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(f′(ξ))qdξ)1q+(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,ν,lγ,δ,k(ωsμ)ds|pvdξ)1p×(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(f′(ξ))qdξ)1q

≤ ‖g‖∞vSv×(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(f′(ξ))qdξ)1q×{(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(ξ−(m(t)ψ(a))pvdξ)1p+(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))−ξ)pvdξ)1p}=2‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))pv+1p×(∫01(f′(m(t)ψ(a)+ξΛ(ψ(b),m(t)ψ(a))))qdξ)1q≤ 2‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))pv+1p×(∫01[m(ξ)h1(ξ)(f′(a))rq+h2(ξ)(f′(b))rq]1rdξ)1q≤ 2‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))pv+1p×[(∫01m1r(ξ)(f′(a))qh11r(ξ)dξ)r+(∫01(f′(b))qh21r(ξ)dξ)r]1rq= 2‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))pv+1p×(f′(a))rqI1r(h1(ξ);m(ξ),r)+(f′(b))rqI2r(h2(ξ);r).rq

So, the proof of this theorem is completed.

Remark 2.9.

In Theorem 2.8, for h1(t)=t,h2(t)=1−t,r=1, if we choose Λ(ψ(b),m(t)ψ(a))=ψ(b)−m(t)ψ(a), where m(t)≡1,∀t∈[0,1] and ψ(x)=x,∀x∈I, then

1. If we put ω=0, we get [[28], Theorem 7].

2. If we put ω=0 along with ν=αk , we get [[10], Theorem 2.5].

3. If we put g(s)=1 and ω=0, we get [[6], Theorem 2.3].

4. If we put ω=0 and ν=1, we get [[6], Corollary 3].

Remark 2.10.

In Theorem 2.8, for h1(t)=t,h2(t)=1−t,r=1, if we choose Λ(ψ(b),m(t)ψ(a))=ψ(b)−m(t)ψ(a), where m(t)≡1,∀t∈[0,1] and ψ(x)=x,∀x∈I, we get [[9], Corollary 3.8].

We point out some special cases of Theorem 2.8.

Corollary 2.11.

In Theorem 2.8 for p=q=2, we get the following inequality:

(2.5)|If,g,E,Λ,ψ,m(ν,a,b)|≤2‖g‖∞νSνΛν+1(ψ(b),m(t)ψ(a))2ν+1(f′(a))2rI1r(h1(ξ);m(ξ),r)+(f′(b))2rI2r(h2(ξ);r)2r.

Corollary 2.12.

In Theorem 2.8 for g(s)=1, we get the following inequality:

(2.6)|If,E,Λ,ψ,m(ν,a,b)|=|(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))Eμ,v,lγ,δ,k(ωsμ)ds)ν[f(m(t)ψ(a))+f(m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a)))]−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫m(t)ψ(a)ξEμ,ν,lγ,δ,k(ωsμ)ds)v−1Eμ,ν,lγ,δ,k(ωξμ)f(ξ)dξ−ν∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))(∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))Eμ,ν,lγ,δ,k(ωsμ)ds)v−1×Eμ,ν,lγ,δ,k(ωsμ)f(ξ)dξ|≤ 2SvΛv+1(ψ(b),m(t)ψ(a))pv+1p(f′(a))rqI1r(h1(ξ);m(ξ),r)+(f′(b))rqI2r(h2(ξ);r).rq

Corollary 2.13.

In Theorem 2.8 for h1(t)=h2(t)=1 and m(t)=m∈(0,1] for all t∈[0,1], we get the following inequality for the generalized relative semi-(m,P)-preinvex mappings: (2.7)

(2.7)|If,g,E,Λ,ψ,m(ν,a,b)|≤2‖g‖∞νSνΛν+1(ψ(b),mψ(a))pν+1pm(f′(a))rq+(f′(b))rqrq.

Corollary 2.14.

In Theorem 2.8 for h1(t)=h(1−t),h2(t)=h(t) and m(t)=m∈(0,1] for all t ∈ [0,1], we get the following inequality for the generalized relative semi-(m,h)-preinvex mappings:

(2.8)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))pν+1p×I2(h(ξ);r)qm(f′(a))rq+(f′(b))rqrq.

Corollary 2.15.

In Corollary 2.14 for h1(t)=(1−t)s and h2(t)=ts, we get the following inequality for the generalized relative semi-(m,s)-Breckner-preinvex mappings:

(2.9)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))pν+1p×rr+sqm(f′(a))rq+(f′(b))rqrq.

Corollary 2.16.

In Corollary 2.14 for h1(t)=(1−t)−s,h2(t)=t−s and 0<s<r, we get the following inequality for the generalized relative semi-(m,s)-Godunova–Levin–Dragomir-preinvex mappings:

(2.10)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))pν+1p×rr−sqm(f′(a))rq+(f′(b))rqrq.

Corollary 2.17.

In Theorem 2.8 for h1(t)=h2(t)=t(1−t) and m(t)=m∈(0,1] for all t∈[0,1], we get the following inequality for the generalized relative semi-(m,tgs)-preinvex mappings:

(2.11)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))pν+1p×β(1+1r,1+1r)qm(f′(a))rq+(f′(b))rqrq.

Corollary 2.18.

In Corollary 2.14 for h1(t)=1−t2t,h2(t)=t21−t and r∈,(12,1] we get the following inequality for the generalized relative semi-m-MT-preinvex mappings:

(2.12)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))pν+1p×β(1−12r,1+12r)qm(f′(a))rq+(f′(b))rqrq.

Theorem 2.19.

Let h1,h2:[0,1]→[0,+∞),ψ:I→K and g:K→ℝ are continuous functions and m:[0,1]→(0,1]. Suppose K=[m(t)ψ(a),m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))]⊆ℝ be an open m-invex subset with respect to Λ:K×K→ℝ for Λ(ψ(b),m(t)ψ(a))>0 and ∀t∈[0,1]. Assume that f:K→(0,+∞) be a differentiable mapping on K∘ such that f′,g∈L(K). If (f′(x))q is the generalized relative semi-m-(r;h1,h2)-preinvex mapping, 0<r≤1,k<l+μ,q≥1 and ‖g‖∞=sups∈K|g(s)|, then the following inequality for ν>0 holds:

(2.13)|If,g,E,Λ,ψ,m(ν,a,b)|≤‖g‖∞νSνΛν+1(ψ(b),m(t)ψ(a))(ν+1)1−1q×{(f′(a))rqI1r(h1(ξ);m(ξ),ν,r)+(f′(b))rqI2r(h2(ξ);ν,r)rq+(f′(a))rqI1¯r(h1(ξ);m(ξ),ν,r)+(f′(b))rqI2¯r(h2(ξ);ν,r)rq},

where

I1(h1(ξ);m(ξ),ν,r):=∫01m1r(ξ)ξνh11r(ξ)dξ; I2(h2(ξ);ν,r):=∫01ξνh21r(ξ)dξ

and

I1¯(h1(ξ);m(ξ),ν,r):=∫01m1r(ξ)(1−ξ)νh11r(ξ)dξ;I2¯(h2(ξ);ν,r):=∫01(1−ξ)νh21r(ξ)dξ.

Proof. From Lemma 2.7, the generalized relative semi-m-(r;h1,h2)-preinvexity of (f′(x))q, the well-known power mean inequality, Minkowski inequality, absolute convergence of Mittag-Leffler function, properties of the modulus, the fact g(s) ≤ ‖g‖∞,∀s ∈ K and changing the variable u=m(t)ψ(a)+ξΛ(ψ(b),m(t)ψ(a)),∀t ∈ [0,1], we have

|If,g,E,Λ,ψ,m(ν,a,b)|≤∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫m(t)ψ(a)ξg(s)Eμ,v,lγ,δ,k(ωsμ)ds|v×|f′(ξ)|dξ+∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,v,lγ,δ,k(ωsμ)ds|v×|f′(ξ)|dξ≤(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫m(t)ψ(a)ξg(s)Eμ,v,lγ,δ,k(ωsμ)ds|v)1−1q×(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫m(t)ψ(a)ξg(s)Eμ,v,lγ,δ,k(ωsμ)ds|v(f′(ξ))qdξ)1q+(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,v,lγ,δ,k(ωsμ)ds|vdξ)1−1q×(∫m(t)ψ(a)m(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))|∫ξm(t)ψ(a)+Λ(ψ(b),m(t)ψ(a))g(s)Eμ,v,lγ,δ,k(ωsμ)ds|v×(f′(ξ))qdξ)1q≤ ‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))(v+1)1−1q×{[∫01ξv(f′(m(t)ψ(a)+ξΛ(ψ(b),m(t)ψ(a))))qdξ]1q+[∫01(1−ξ)v(f′(m(t)ψ(a)+ξΛ(ψ(b),m(t)ψ(a))))qdξ]1q}≤ ‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))(v+1)1−1q

×{[∫01ξv[m(ξ)h1(f′(a))rq+h2(ξ)(f′(a))rq]1rdξ]1q+[∫01(1−ξ)v[m(ξ)h1(f′(a))rq+h2(ξ)(f′(a))rq]1rdξ]1q}≤ ‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))(v+1)1−1q×{[(∫01m1r(ξ)(f′(a))qξvh11r(ξ)dξ)r+(∫01(f′(b))qξvh21r(ξ)dξ)r]1rq+[(∫01m1r(ξ)(f′(a))q(1−ξ)vh11r(ξ)dξ)r+(∫01(f′(b))q(1−ξ)vh21r(ξ)dξ)r]1rq}=‖g‖∞vSvΛv+1(ψ(b),m(t)ψ(a))(v+1)1−1q×{(f′(a))rqI1r(h1(ξ);m(ξ),v,r)+(f′(a))rqI2r(h2(ξ);v,r)rq+(f′(a))rqI1¯r(h1(ξ);m(ξ),v,r)+(f′(a))rqI2¯r(h2(ξ);v,r)rq}.

So, the proof of this theorem is completed.

We point out some special cases of Theorem 2.19.

Corollary 2.20.

In Theorem 2.19 for q=1, we get the following inequality:

(2.14)|If,g,E,Λ,ψ,m(ν,a,b)|≤‖g‖∞νSνΛν+1(ψ(b),m(t)ψ(a))×{(f′(a))rI1r(h1(ξ);m(ξ),ν,r)+(f′(b))rI2r(h2(ξ);ν,r)r+(f′(a))rI1¯r(h1(ξ);m(ξ),ν,r)+(f′(b))rI2¯r(h2(ξ);ν,r)r}.

Corollary 2.21.

In Theorem 2.19 for g(s)=1, we get the following inequality:

(2.15)|If,E,Λ,ψ,m(ν,a,b)|≤SνΛν+1(ψ(b),m(t)ψ(a))(ν+1)1−1q×{(f′(a))rqI1r(h1(ξ);m(ξ),ν,r)+(f′(b))rqI2r(h2(ξ);ν,r)rq+(f′(a))rqI1¯r(h1(ξ);m(ξ),ν,r)+(f′(b))rqI2¯r(h2(ξ);ν,r)rq}.

Corollary 2.22.

In Theorem 2.19 for h1(t)=h2(t)=1 and m(t)=m∈(0,1] for all t∈[0,1], we get the following inequality for the generalized relative semi-(m,P)-preinvex mappings:

(2.16)|If, g, E, Λ, ψ, m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))ν+1m(f′(a))rq+(f′(b))rqrq.

Corollary 2.23.

In Theorem 2.19 for h1(t)=h(1−t),h2(t)=h(t) and m(t)=m∈(0,1] for all t∈[0,1], we get the following inequality for the generalized relative semi- (m,h)-preinvex mappings:

(2.17)|If, g, E, Λ, ψ, m(ν,a,b)| ≤ ‖g‖∞νSνΛν+1(ψ(b),mψ(a))(ν+1)1−1q×{m(f′(a))rqI2r(h(1−ξ);ν,r)+((f′(b))rqI2r(h(ξ);ν,r)rq+m(f′(a))rqI1¯r(h(1−ξ);ν,r)+((f′(b))rqI2¯r(h(ξ);ν,r)rq}.

Corollary 2.24.

In Corollary 2.23 for h1(t)=(1−t)s and h2(t)=ts, we get the following inequality for the generalized relative semi-(m,s)-Breckner-preinvex mappings:

(2.18)|If,g,E,Λ,ψ,m(ν,a,b)|≤‖g‖∞νSνΛν+1(ψ(b),mψ(a))(ν+1)1−1q×{m(f′(a))rqβr(sr+1,v+1)+(f′(b))rq(1sr+ν+1)rrq+m(f′(a))rq(1sr+v+1)r+(f′(b))rqβr(sr+1,v+1)rq}.

Corollary 2.25.

In Corollary 2.23 for h1(t)=(1−t)−s,h2(t)=t−s and 0<s<r, we get the following inequality for the generalized relative semi-(m,s)-Godunova–Levin–Dragomir-preinvex mappings:

(2.19)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ ‖g‖∞νSνΛν+1(ψ(b),mψ(a))(v+1)1−1q×{m(f′(a))rqβr(1−sr,ν+1)+(f′(b))rq(1ν−sr+1)rrq+m(f′(a))rq(1v−st+1)r+(f′(b))rqβr(1−sr,v+1)rq}.

Corollary 2.26.

In Theorem 2.19 for h1(t)=h2(t)=t(1−t) and m(t)=m∈(0,1] for all t∈[0,1], we get the following inequality for the generalized relative semi-(m,tgs)-preinvex mappings:

(2.20)|If,g,E,Λ,ψ,m(ν,a,b)| ≤ 2‖g‖∞νSνΛν+1(ψ(b),mψ(a))(ν+1)1−1qβ(1+1r,ν+1r+1)q×m(f′(a))rq+(f′(b))rqrq.

Corollary 2.27.

In Corollary 2.23 for h1(t)=1−t2t,h2(t)=t21−t and r∈(12,1], we get the following inequality for the generalized relative semi-m- MT-preinvex mappings:

(2.21)|If,g,E,Λ,ψ,m(ν,a,b)|≤‖g‖∞νSνΛν+1(ψ(b),mψ(a))(v+1)1−1q×{m(f′(a))rqβr(ν−12r+1,1+12r)+(f′(b))rqβr(ν+12r+1,1−12r)rq+m(f′(a))rqβr(ν+12r+1,1−12r)+(f′(b))rqβr(ν−12r+1,1+12r)rq}.

Remark 2.28.

By taking particular values of parameters used in Mittag-Leffler function in Theorems 2.8 and 2.19, several fractional integral inequalities can be obtained.

Remark 2.29.

Also, applying our Theorems 2.8 and 2.19, for f′(x) ≤ K, for all x∈I, we can get some new fractional integral inequalities.

3. Applications to special means

Definition 3.1.

([2]). A function M:ℝ+2→ℝ+, is called a Mean function if it has the following properties:

1. Homogeneity: M(ax,ay)=aM(x,y), for all a>0,

2. Symmetry: M(x,y)=M(y,x),

3. Reflexivity: M(x,x)=x,

4. Monotonicity: If x≤x′ and y≤y′, then M(x,y)≤M(x′,y′),

5. Internality: m{x,y}≤M(x,y)≤m{x,y}.

Let us consider some special means for arbitrary positive real numbers α≠β as follows: The arithmetic mean A:=A(α,β); The geometric mean G:=G(α,β); The harmonic mean H:=H(α,β); The power mean Pr:=Pr(α,β); The identric mean I:=I(α,β); The logarithmic mean L:=L(α,β); The generalized log-mean Lp:=Lp(α,β); The weighted p-power mean M=Mp. Now, let a and b be positive real numbers such that a<b. Consider the function M¯:=M(ψ(a),ψ(b)):[ψ(a),ψ(a)+Λ(ψ(b),ψ(a))]×[ψ(a),ψ(a)+Λ(ψ(b),ψ(a))]→ℝ+, which is one of the above mentioned means, therefore one can obtain various inequalities using the results of Section 2 for these means as follows: Replace Λ(ψ(y),m(t)ψ(x)) with Λ(ψ(y),ψ(x)) where m(t) ≡ 1, for all t ∈ [0,1] and setting Λ(ψ(y),ψ(x))=M(ψ(x),ψ(y)) for all x,y∈I, in (2.4) and (2.13), one can obtain the following interesting inequalities involving means:

(3.1)|If, g, E, M¯, ψ(ν,a,b)| ≤ 2‖g‖∞νSνM¯ν+1pv+1p×(f′(a))rqI2r(h1(ξ);r)+(f′(b))rqI2r(h2(ξ);r)rq,

(3.2)|If, g, E, M¯, ψ(ν,a,b)|≤‖g‖∞νSvM¯ν+1(v+1)1−1q×{(f′(a))rqI2r(h1(ξ);ν,r)+(f′(b))rqI2r(h2(ξ);ν,r)rq+(f′(a))rqI1¯r(h1(ξ);ν,r)+(f′(b))rqI2¯r(h2(ξ);ν,r)rq}.

Letting M¯:=A,G,H,Pr,I,L,Lp,Mp in (3.1) and (3.2), we get the inequalities involving means for particular choices of (f′(x))q that are the generalized relative semi-1-(r;h1,h2)-preinvex mappings.

Remark 3.2.

Also, applying our Theorems 2.8 and 2.19 for appropriate choices of functions h1 and h2 (see Remark 2.6) such that (f′(x))q to be the generalized relative semi-1-(r;h1,h2)-preinvex mappings (see examples: f(x)=xα, where α>1,∀x>0; f(x)=1x,∀x>0; f(x)=ex,∀x∈ℝ; f(x)=−lnx,∀x>0; etc.), we can deduce some new inequalities using above special means. The details are left to the interested reader.