Abstract
In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals with exponential kernels for convex functions. Finally, some applications to special means of real number are presented.
Highlights
Let h : [ a, b] ⊂ < → < be a convex function.Hermite–Hadamard inequality h a+b ≤ b−a Z b a h(s)ds ≤h meets the following classic h( a) + h(b) (1)If h is a concave function, the inequalities in (1) are presented in the negative direction.The Hermite–Hadamard inequality provides us the estimates for the integral average of a continuous convex function on a compact interval.For the latest results on generalizing, improving, and extending this classical Hermite–Hadamard inequality, one can see [2,3,4,5,6,7,8,9] and the references therein
Based on the above interpretation, we acquire the bound estimates of the difference between the average of the fractional integrals with an exponential kernel and the mean values of the endpoints and the midpoint
With the same order derivatives of a given function, the Hermite–Hadamard-type inequalities involving different fractional integrals tend to be same when α → 1
Summary
Let h : [ a, b] ⊂ < → < be a convex function. If h is a concave function, the inequalities in (1) are presented in the negative direction. If |h00 | is convex, it is natural to study the right- and left-type Hermite–Hadamard inequality via the fractional integral with an exponential kernel similar to Lemma 2, i.e., we want to find the constants ρ1 and ρ2 satisfying the following inequities:. Motivated by [12,15,16], we will demonstrate three new fractional-type integral identities and set up their corresponding Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals for convex functions, respectively. If h : [ a, b] ⊆ < → < is differentiable, |h0 | is convex on [a,b], and h0 ∈ L[ a, b], the following inequality about the fractional integrals (4) and (5) holds:. By (14), we will prove the Hermite–Hadamard-type inequality of the order derivatives via the fractional integrals with an exponential kernel for convex functions. S2 |h00 ( a)| + t(1 − t)|h00 (b)| ds + 1 s(1 − s)|h00 ( a)|ds + (1 − s)2 |h00 (b)| ds e s ds ρ (1 − e − ρ ) 0
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