Fractional integral inequalities of Hermite-Hadamard type for convex functions with respect to a monotone function
aa r X i v : . [ m a t h . G M ] M a y Fractional integral inequalities of Hermite-Hadamard type forconvex functions with respect to a monotone function
Pshtiwan Othman Mohammed ∗ Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq
Abstract
In the literature, the left-side of Hermite–Hadamard’s inequality is called a midpoint type inequality.In this article, we obtain new integral inequalities of midpoint type for Riemann–Liouville fractional inte-grals of convex functions with respect to increasing functions. The resulting inequalities generalize somerecent classical integral inequalities and Riemann–Liouville fractional integral inequalities established inearlier works. Finally, applications of our work are demonstrated via the known special functions of realnumbers.
A function g : I ⊆ R → R is said to be convex on the interval I , if the inequality g ( η x + (1 − η ) y ) ≤ η g ( x ) + (1 − η ) g ( y ) (1.1)holds for all x, y ∈ I and η ∈ [0 , g is concave, provided − g is convex.For convex functions (1.1), many equalities and inequalities have been established, e.g. , Ostrowski typeinequality [1], Opial inequality [2], Hardy type inequality [3], Olsen type inequality [4], Gagliardo-Nirenbergtype inequality [5], midpoint and trapezoidal type inequalities [6, 7] and the Hermite–Hadamard type (HH-type) inequality [8] that will be used in our study, which is defined by: g (cid:18) u + v (cid:19) ≤ v − u Z vu g ( x ) dx ≤ g ( u ) + g ( v )2 , (1.2)where g : I ⊆ R → R is assumed to be a convex function on I where a, b ∈ I with u < v .A huge number of researchers in the field of applied and pure mathematics have devoted their efforts tomodify, generalize, refine, and extend the Hermite–Hadamard inequality (1.2) for convex and other classesof convex functions; see for further details [8–12].In 2013, the HH-type inequality (1.2) has been generalised to fractional integrals of Riemann–Liouvilletype by Sarikaya et al [13]. Their result is as follows, for an L convex function f : [ u, v ] → R , and for any µ > g (cid:18) u + v (cid:19) ≤ Γ( µ + 2)2( v − u ) µ (cid:2) I µu + g ( v ) + I µv − g ( u ) (cid:3) ≤ g ( u ) + g ( v )2 , (1.3)where I µu + and I µv − denote left-sided and right-sided Riemann-Liouville fractional integrals of order µ > I µu + g ( x ) = 1Γ( µ ) Z xu ( x − t ) µ − g ( t ) dt, x > u,I µv − g ( x ) = 1Γ( µ ) Z vx ( t − x ) µ − g ( t ) dt, x < v. (1.4) Keywords:
Fractional integral; Convex functions; Hermite-Hadamard inequality
Mathematics subject classification (2010): ∗ Corresponding author. Email: [email protected]
1f we take µ = 1 in (1.3) we obtain (1.2), it is clear that inequality (1.3) is a generalization of Hermite–Hadamard inequality (1.2). Many further results have been derived from this [15–20], including in differenttypes of fractional calculus, e.g. for tempered fractional integrals [21], those of Hilfer type [22], for thosemodels of fractional calculus involving Mittag-Leffler kernels [23], and for fractional integrals with respectto functions [24]. But so far such inequalities have not been investigated for fractional integrals of a twicedifferentiable convex function with respect to a monotone function. For this reason, we recall the Riemann–Liouville fractional integrals of a function with respect to a monotone function. Definition 1.1.
Let ( u, v ) ⊆ ( −∞ , ∞ ) be a finite or infinite interval of the real-axis R and µ >
0. Let ψ ( x )be an increasing and positive monotone function on the interval ( u, v ] with a continuous derivative ψ ′ ( x ) onthe interval ( u, v ). Then the left and right-sided ψ -Riemann–Liouville fractional integrals of a function g with respect to another function ψ ( x ) on [ u, v ] are defined by [14, 25, 26]: I µ : ψu + g ( x ) = 1Γ( µ ) Z xu ψ ′ ( t )( ψ ( x ) − ψ ( t )) µ − g ( t ) dt,I µ : ψv − g ( x ) = 1Γ( µ ) Z vx ψ ′ ( t )( ψ ( t ) − ψ ( x )) µ − g ( t ) dt. (1.5)It is important to note that if we set ψ ( x ) = x in (1.5), then ψ -Riemann–Liouville fractional integral reducesto Riemann–Liouville fractional integral (1.4).As we said, in this study we investigate several inequalities of midpoint type for Riemann–Liouvillefractional integrals of twice differentiable convex functions with respect to increasing functions. Our main results follow the following lemma:
Lemma 2.1.
Let g : [ u, v ] ⊆ R → R be a differentiable function and g ′′ ∈ L [ u, v ] with ≤ u < v . If ψ ( x ) isan increasing and positive monotone function on ( u, v ] and its derivative ψ ′ ( x ) is continuous on ( u, v ) , thenfor µ ∈ (0 , we have σ µ,ψ ( g ; u, v ) = 2 µ − ( v − u ) µ "Z ψ − ( v ) ψ − ( u + v ) ψ ′ ( t )( v − ψ ( t )) µ +1 ( g ′′ ◦ ψ )( t ) dt − Z ψ − ( u + v ) ψ − ( u ) ψ ′ ( t )( ψ ( t ) − u ) µ +1 ( g ′′ ◦ ψ )( t ) dt , (2.1) where σ µ,ψ ( g ; u, v ) = 2 µ − Γ( µ + 2)( v − u ) µ (cid:20) I µ : ψψ − ( u + v ) + ( g ◦ ψ ) (cid:0) ψ − ( v ) (cid:1) + I µ : ψψ − ( u + v ) − ( g ◦ ψ ) (cid:0) ψ − ( u ) (cid:1)(cid:21) − ( µ + 1) g (cid:18) u + v (cid:19) . Proof.
From Definition 1.1 we have ~ := 2 µ − Γ( µ + 2)( v − u ) µ I µ : ψψ − ( u + v ) + ( g ◦ ψ ) (cid:0) ψ − ( v ) (cid:1) = µ ( µ + 1)2 µ − ( v − u ) µ Z ψ − ( v ) ψ − ( u + v ) ψ ′ ( t )( v − ψ ( t )) µ − ( g ◦ ψ )( t ) dt = − ( µ + 1)2 µ − ( v − u ) µ Z ψ − ( v ) ψ − ( u + v )( g ◦ ψ )( t ) d ( v − ψ ( t )) µ . Integrating by parts twice, we have ~ = µ + 12 g (cid:18) u + v (cid:19) + ( µ + 1)2 µ − ( v − u ) µ Z ψ − ( v ) ψ − ( u + v ) ψ ′ ( t )( v − ψ ( t )) µ ( g ′ ◦ ψ )( t ) dt = µ + 12 g (cid:18) u + v (cid:19) + 12 g ′ (cid:18) u + v (cid:19) + ( µ + 1)2 µ − ( v − u ) µ Z ψ − ( v ) ψ − ( u + v ) ψ ′ ( t )( v − ψ ( t )) µ +1 ( g ′′ ◦ ψ )( t ) dt (2.2)2nalogously ~ := 2 µ − Γ( µ + 1)( v − u ) µ I µ : ψψ − ( u + v ) + ( g ◦ ψ ) (cid:0) ψ − ( v ) (cid:1) = µ + 12 g (cid:18) u + v (cid:19) − g ′ (cid:18) u + v (cid:19) − µ − ( v − u ) µ Z ψ − ( u + v ) ψ − ( u ) ψ ′ ( t )( ψ ( t ) − u ) µ +1 ( g ′′ ◦ ψ )( t ) dt. (2.3)It follows from (2.2) and (2.3) that ~ + ~ − ( µ + 1) g (cid:18) u + v (cid:19) = 2 µ − ( v − u ) µ "Z ψ − ( v ) ψ − ( u + v ) ψ ′ ( t )( v − ψ ( t )) µ +1 ( g ′′ ◦ ψ )( t ) dt − Z ψ − ( u + v ) ψ − ( u ) ψ ′ ( t )( ψ ( t ) − u ) µ +1 ( g ′′ ◦ ψ )( t ) dt . This completes the proof of Lemma 2.1.
Corollary 2.1.
With the similar assumptions of Lemma 2.1 if1. ψ ( x ) = x , we have µ − Γ( µ + 2)( v − u ) µ (cid:20) I µ ( u + v ) + g ( v ) + I µ ( u + v ) − g ( u ) (cid:21) − ( µ + 1) g (cid:18) u + v (cid:19) = ( v − u ) "Z t µ +1 g ′′ (cid:18) t u + 2 − t v (cid:19) dt + Z t µ +1 g ′′ (cid:18) − t u + t v (cid:19) dt , which is obtained by Tomar et al. [27].2. ψ ( x ) = x and µ = 1 , we have v − u Z vu g ( x ) dx − g (cid:18) u + v (cid:19) = ( v − u ) "Z t g ′′ (cid:18) t u + 2 − t v (cid:19) dt + Z t g ′′ (cid:18) − t u + t v (cid:19) dt , which is obtained by Sarikaya and Kiris [28]. Theorem 2.1.
Let g : [ u, v ] ⊆ R → R be a differentiable function and g ′′ ∈ L [ u, v ] with ≤ u < v . Supposethat | g ′′ | is convex on [ u, v ] , ψ ( x ) is an increasing and positive monotone function on ( u, v ] and its derivative ψ ′ ( x ) is continuous on ( u, v ) , then for µ ∈ (0 , we have | σ µ,ψ ( g ; u, v ) | ≤ ( v − u ) (cid:18) µ + 2 (cid:19) − q ((cid:20) µ + 3) | g ′′ ( u ) | q + (cid:18) µ + 2 − µ + 3) (cid:19) | g ′′ ( v ) | q (cid:21) q + (cid:20)(cid:18) µ + 2 − µ + 3) (cid:19) | g ′′ ( u ) | q + 12( µ + 3) | g ′′ ( v ) | q (cid:21) q ) (2.4) for q ≥ .Proof. Suppose that q = 1. By means of Lemma 2.1 and Definition 1.1, we get σ µ,ψ ( g ; u, v ) = 2 µ − ( v − u ) µ "Z ψ − ( v ) ψ − ( u + v ) ψ ′ ( t )( v − ψ ( t )) µ +1 ( g ′′ ◦ ψ )( t ) dt − Z ψ − ( u + v ) ψ − ( u ) ψ ′ ( t )( ψ ( t ) − u ) µ +1 ( g ′′ ◦ ψ )( t ) dt . (2.5)3hange the variables x = v − ψ ( t )) v − u and x = ψ ( t ) − u ) v − u and then set t = x = x into the resulting equality,then (2.5) becomes σ µ,ψ ( g ; u, v ) = ( v − u ) "Z t µ +1 g ′′ (cid:18) t u + 2 − t v (cid:19) dt + Z t µ +1 g ′′ (cid:18) − t u + t v (cid:19) dt , that is | σ µ,ψ ( g ; u, v ) | ≤ ( v − u ) "Z t µ +1 (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) − t u + t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) dt . (2.6)By using the convexity of | g ′′ | , then inequality (2.6) gives | σ µ,ψ ( g ; u, v ) | ≤ ( v − u ) "Z t µ +1 (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) − t u + t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) dt ≤ ( v − u ) (cid:18) | g ′′ ( u ) | Z t µ +2 dt + | g ′′ ( v ) | Z − t t µ +1 dt + | g ′′ ( v ) | Z t µ +2 dt + | g ′′ ( u ) | Z − t t µ +1 dt (cid:19) = ( v − u ) µ + 2) ( | g ′′ ( u ) | + | g ′′ ( v ) | ) . This gives (2.4) for q = 1.Now, suppose that q >
1. Using inequality of (2.6), convexity of | g ′′ | q and the power–mean’s inequalityfor q >
1, we have Z t µ +1 (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z t µ +1 − µ +1 q (cid:20) t µ +1 q (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) dt ≤ (cid:18)Z t µ +1 (cid:19) − q (cid:18)Z t µ +1 (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt (cid:19) q ≤ (cid:18) µ + 2 (cid:19) − q (cid:18)Z (cid:18) t µ +2 | g ′′ ( u ) | q + 2 t µ +1 − t µ +2 | g ′′ ( v ) | q (cid:19) dt (cid:19) q = (cid:18) µ + 2 (cid:19) − q (cid:20) µ + 3) | g ′′ ( u ) | q + (cid:18) µ + 2 − µ + 3) (cid:19) | g ′′ ( v ) | q (cid:21) q . (2.7)In the same manner, we get Z t µ +1 (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) − t u + t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:18) µ + 2 (cid:19) − q (cid:20)(cid:18) µ + 2 − µ + 3) (cid:19) | g ′′ ( u ) | q + 12( µ + 3) | g ′′ ( v ) | q (cid:21) q . (2.8)Using (2.7) and (2.8) in (2.6) we obtain (2.4) for q >
1. Thus the proof of theorem 2.1 is completed.
Corollary 2.2.
With the similar assumptions of Theorem 2.1 if1. ψ ( x ) = x , we have (cid:12)(cid:12)(cid:12)(cid:12) µ − Γ( µ + 2)( v − u ) µ (cid:20) I µ ( u + v ) + g ( v ) + I µ ( u + v ) − g ( u ) (cid:21) − ( µ + 1) g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) (cid:18) µ + 2 (cid:19) − q ((cid:20) µ + 3) | g ′′ ( u ) | q + (cid:18) µ + 2 − µ + 3) (cid:19) | g ′′ ( v ) | q (cid:21) q + (cid:20)(cid:18) µ + 2 − µ + 3) (cid:19) | g ′′ ( u ) | q + 12( µ + 3) | g ′′ ( v ) | q (cid:21) q ) , which is obtained by Tomar et al. [27]. . ψ ( x ) = x and µ = 1 , we have (cid:12)(cid:12)(cid:12)(cid:12) v − u Z vu g ( x ) dx − g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) "(cid:18) | g ′′ ( u ) | q + 5 | g ′′ ( v ) | q (cid:19) q + (cid:18) | g ′′ ( u ) | q + 3 | g ′′ ( v ) | q (cid:19) q , which is obtained by Sarikaya et al. [29].3. ψ ( x ) = x and q = 1 , we have (cid:12)(cid:12)(cid:12)(cid:12) µ − Γ( µ + 2)( v − u ) µ (cid:20) I µ ( u + v ) + g ( v ) + I µ ( u + v ) − g ( u ) (cid:21) − ( µ + 1) g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) µ + 2) (cid:18) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:19) , which is obtained by Tomar et al. [27].4. ψ ( x ) = x, µ = 1 and q = 1 , we have (cid:12)(cid:12)(cid:12)(cid:12) v − u Z vu g ( x ) dx − g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) (cid:18) | g ′′ ( u ) | + | g ′′ ( v )2 (cid:19) , which is obtained by Sarikaya et al. [29]. Theorem 2.2.
Let g : [ u, v ] ⊆ R → R be a differentiable function and g ′′ ∈ L [ u, v ] with ≤ u < v . Supposethat | g ′′ | q is convex on [ u, v ] , ψ ( x ) is an increasing and positive monotone function on ( u, v ] and its derivative ψ ′ ( x ) is continuous on ( u, v ) , then for µ ∈ (0 , we have | σ µ,ψ ( g ; u, v ) | ≤ ( v − u ) (cid:18) µ + 1) p + 1 (cid:19) p "(cid:18) | g ′′ ( u ) | q + 3 | g ′′ ( v ) | q (cid:19) q + (cid:18) | g ′′ ( u ) | q + | g ′′ ( v ) | q (cid:19) q ≤ ( v − u ) (cid:18) µ + 1) p + 1 (cid:19) p (cid:0) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:1) , (2.9) such that q > and p + q = 1 .Proof. By using the Holder’s inequality, we have Z t µ +1 (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:18)Z t ( µ +1) p (cid:19) p (cid:18)Z (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) t u + 2 − t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q dt (cid:19) q ≤ (cid:18) µ + 1) p + 1 (cid:19) p (cid:18)Z (cid:18) t | g ′′ ( u ) | q + 2 − t | g ′′ ( v ) | q (cid:19) dt (cid:19) q = (cid:18) µ + 1) p + 1 (cid:19) p (cid:18) | g ′′ ( u ) | q + 3 | g ′′ ( v ) | q (cid:19) q . (2.10)Similarly, we have Z t µ +1 (cid:12)(cid:12)(cid:12)(cid:12) g ′′ (cid:18) − t u + t v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:18) µ + 1) p + 1 (cid:19) p (cid:18) | g ′′ ( u ) | q + | g ′′ ( v ) | q (cid:19) q . (2.11)Thus, the inequalities (2.6), (2.10) and (2.11) complete the proof of the first inequality of (2.9).To prove the second inequality of (2.9), we apply the formula n X i =1 ( c i + d i ) m ≤ n X i =1 c mi + n X i =1 + d mi , ≤ m < c = 3 | g ′′ ( u ) | q , c = | g ′′ ( u ) | q , d = | g ′′ ( v ) | q , d = 3 | g ′′ ( v ) | q and m = q . Then (2.6) gives | σ µ,ψ ( g ; u, v ) | ≤ ( v − u ) (cid:18) µ + 1) p + 1 (cid:19) p "(cid:18) | g ′′ ( u ) | q + 3 | g ′′ ( v ) | q (cid:19) q + (cid:18) | g ′′ ( u ) | q + | g ′′ ( v ) | q (cid:19) q ≤ ( v − u ) (cid:16) q + 1 (cid:17) (cid:18) µ + 1) p + 1 (cid:19) p (cid:2) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:3) ≤ ( v − u ) (cid:18) µ + 1) p + 1 (cid:19) p (cid:0) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:1) . Hence the proof of Theorem 2.2 is completed.
Corollary 2.3.
With the similar assumptions of Theorem 2.2, if1. ψ ( x ) = x , we have (cid:12)(cid:12)(cid:12)(cid:12) µ − Γ( µ + 2)( v − u ) µ (cid:20) I µ ( u + v ) + g ( v ) + I µ ( u + v ) − g ( u ) (cid:21) − ( µ + 1) g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) (cid:18) µ + 1) p + 1 (cid:19) p "(cid:18) | g ′′ ( u ) | q + 3 | g ′′ ( v ) | q (cid:19) q + (cid:18) | g ′′ ( u ) | q + | g ′′ ( v ) | q (cid:19) q ≤ ( v − u ) (cid:18) µ + 1) p + 1 (cid:19) p (cid:0) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:1) , which is obtained by Tomar et al. [27].2. ψ ( x ) = x and µ = 1 , we have (cid:12)(cid:12)(cid:12)(cid:12) v − u Z vu g ( x ) dx − g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) p + 1) p "(cid:18) | g ′′ ( u ) | q + 3 | g ′′ ( v ) | q (cid:19) q + (cid:18) | g ′′ ( u ) | q + | g ′′ ( v ) | q (cid:19) q ≤ ( v − u ) q (2 p + 1) p (cid:0) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:1) , which is obtained by Sarikaya et al. [29]. Corollary 2.4.
From Theorems 2.1–2.2, we obtain the following inequality for ψ ( x ) = x, µ = 1 and q > : (cid:12)(cid:12)(cid:12)(cid:12) v − u Z vu g ( x ) dx − g (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) min { δ , δ } (cid:0) | g ′′ ( u ) | + | g ′′ ( v ) | (cid:1) , where δ = and δ =
2+ 2 q (2 p +1) p such that p = qq − . In this section some applications are presented to demonstrate usefulness of our obtained results in theprevious sections.
Let u and v be two arbitrary positive real numbers, then consider the following special means:(i) The arithmetic mean: A = A ( u, v ) = u + v . H = H ( u, v ) = 2 u + v , u, v = 0 . (iii) The geometric mean: G = G ( u, v ) = √ u v. (iv) The logarithmic mean: L ( u, v ) = v − u log( v ) − log( u ) , u = v. (v) The generalized logarithmic mean: L n ( u, v ) = (cid:20) v n +1 − u n +1 ( v − u )( n + 1) (cid:21) n , n ∈ Z \ {− , } . Proposition 3.1.
Let | n | ≥ and u, v ∈ R with < u < v , then | A n ( u, v ) − L nn ( u, v ) | ≤ ( v − u ) | n ( n − | · q +2 h A q (cid:16) | u | ( n − q , | v | ( n − q (cid:17) + A q (cid:16) | u | ( n − q , | v | ( n − q (cid:17)i , (3.1) for q ≥ .Proof. Apply Corollary 2.2 part (2) for g ( x ) = x n , where n as specified above. Proposition 3.2.
Let u, v ∈ R with < u < v , then (cid:12)(cid:12) A − ( u, v ) − L − ( u, v ) (cid:12)(cid:12) ≤ ( v − u ) · q +2 h A q (cid:0) | u | − q , | v | − q (cid:1) + A q (cid:0) | u | − q , | v | − q (cid:1)i , (3.2) for q ≥ .Proof. Apply Corollary 2.2 part (2) for g ( x ) = x , x = 0. Proposition 3.3.
Let | n | ≥ and u, v ∈ R with < u < v , then (cid:12)(cid:12) H − n ( v, u ) − L nn (cid:0) v − , u − (cid:1)(cid:12)(cid:12) ≤ (cid:0) v − − u − (cid:1) | n ( n − | · q +2 h H − q (cid:16) | u | ( n − q , | v | ( n − q (cid:17) + H − q (cid:16) | u | ( n − q , | v | ( n − q (cid:17)i , (3.3) and (cid:12)(cid:12) H ( v, u ) − L − (cid:0) v − , u − (cid:1)(cid:12)(cid:12) ≤ (cid:0) v − − u − (cid:1) · q +2 h H − q (cid:0) | u | − q , | v | − q (cid:1) + H − q (cid:0) | u | − q , | v | − q (cid:1)i , (3.4) for q ≥ .Proof. Observe that A − (cid:0) u − , v − (cid:1) = H ( u, v ) = u + v . So, Make the change of variables u → v − and v → u − in the inequalities (3.1) and (3.2), we can deduce the desired inequalities (3.3) and (3.4) respectively. Proposition 3.4.
Let u, v ∈ R with < u < v , then (cid:12)(cid:12) G − ( u, v ) − A − ( u, v ) (cid:12)(cid:12) ≤ ( b − a ) · q +1 h A q (cid:0) | u | − q , | v | − q (cid:1) + A q (cid:0) | u | − q , | v | − q (cid:1)i , (3.5) for q ≥ .Proof. Apply Corollary 2.2 part (2) for g ( x ) = x . 7ow, we give an application to a midpoint formula. Let d be a partition u = x < x < · · · < x m − Let g : [ u, v ] → R be a differentiable mapping on ( u, v ) with u < v . Suppose that | g ′′ | q , q ≥ be a convex function, then for every partition of [ u, v ] the midpoint error satisfies | E ( g, d ) | ≤ min { δ , δ } m − X j =0 ( x j +1 − x j ) (cid:0) | g ′′ ( x j ) | + | g ′′ ( x j +1 ) | (cid:1) . (3.6) Proof. From Corollary 2.4, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x j +1 x j g ( x ) dx − ( x j +1 − x j ) g (cid:18) x j + x j +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ min { δ , δ } ( x j +1 − x j ) (cid:0) | g ′′ ( x j ) | + | g ′′ ( x j +1 ) | (cid:1) Summing over j from 0 to m − | g ′′ | is convex, we obtain, by the triangleinequality, that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba g ( x ) dx − T ( g, d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m − X j =0 "Z x j +1 x j g ( x ) dx − ( x j +1 − x j ) g (cid:18) x j + x j +1 (cid:19) ≤ m − X j =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x j +1 x j g ( x ) dx − ( x j +1 − x j ) g (cid:18) x j + x j +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ min { δ , δ } m − X j =0 ( x j +1 − x j ) (cid:0) | g ′′ ( x j ) | + | g ′′ ( x j +1 ) | (cid:1) . This ends the proof. Let the function I p : R → [1 , ∞ ) be defined by I p ( x ) = 2 p Γ( p + 1) x − v I p ( x ) , x ∈ R . For this we recall the modified Bessel function of the first kind I p which is defined as [30]: I p ( x ) = X n ≥ (cid:0) x (cid:1) p +2 n n !Γ( p + n + 1) . The first and the n th order derivative formula of I p ( x ) are, respectively, given by [31]: I ′ p ( x ) = x p + 1) I p +1 ( x ) , (3.7) ∂ n I p ( x ) ∂x n = 2 n − p √ πx p − n Γ( p + 1) F (cid:18) p + 12 , p + 22 ; p + 1 − n , p + 2 − n , p + 1; x (cid:19) , (3.8)8here F ( · , · ; · , · , · ; · ) is the hypergeometric function defined by [31]: F (cid:18) p + 12 , p + 22 ; p + 1 − n , p + 2 − n , p + 1; x (cid:19) = ∞ X k =0 (cid:0) p +12 (cid:1) k (cid:0) p +22 (cid:1) k (cid:0) p − (cid:1) k (cid:0) p − (cid:1) k ( p + 1) k x k k ( k )! , (3.9)where, for some parameter ν , the Pochhammer symbol ( ν ) k is defined as( ν ) = 1 , ( ν ) k = ν ( ν + 1) · · · ( ν + k − , k = 1 , , ... Proposition 3.6. Let u, v ∈ R with < u < v , then for each p > − we have (cid:12)(cid:12)(cid:12)(cid:12) I p ( v ) − I p ( u ) v − u − a + b p + 1) I p +1 (cid:18) u + v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( v − u ) min { δ , δ } − p √ π Γ( p + 1) × | a | p − (cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) p + 12 , p + 22 ; p + 1 − n , p + 2 − n , p + 1; a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | b | p − (cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) p + 12 , p + 22 ; p + 1 − n , p + 2 − n , p + 1; b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)! . (3.10) Proof. Let g ( x ) = I ′ p ( x ). Note that the function x 7→ I ′′′ p ( x ) is convex on the interval [0 , ∞ ) for each p > − In this paper, we established some new integral inequalities of midpoint type for convex functions with respectto increasing functions involving Riemann–Liouville fractional integrals. It can be noted from Corollary 2.1–2.3 that our results are a generalization of all obtained results in [27–29]. References [1] B. Gavrea, I. 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