On some Hermite–Hadamard inequalities involving k-fractional operators

The main objective of this paper is to establish some new Hermite–Hadamard type inequalities involving k-Riemann–Liouville fractional integrals. Using the convexity of differentiable functions some related inequalities have been proved, which have deep connection with some known results. At the end, some applications of the obtained results in error estimations of quadrature formulas are also considered.


Introduction
In literature, inequalities are very important for convex functions especially the integral inequalities for convex functions originated form Hermite and Hadamard (see [11, p. 137]).
The researchers have worked on Hermite-Hadamard type inequalities since 1893 [4]. The classical Hermite-Hadamard inequality reads as follows: if f : I → R is convex on the interval I of real numbers and a, b ∈ I with a < b, then We note that the Hermite-Hadamard inequality may be assessed as a treatment of the conception of convexity. The Hermite-Hadamard inequality for convex functions has conferred revived awareness in the latest years and some unusual variations of essential and conclusion have been established (see, for example, [5,6,14,17]). In the last few years, the theory of inequalities has progressed very fast. The evolution of the hypothesis associated with ancient inequalities has developed in a resumption of attentiveness in this field.
In many classical inequalities, the Hermite-Hadamard inequality is one of the important inequality of analysis. Such an inequality has been applied for different types of problems of fractional calculus (see [1, 2, 8-10, 15, 16]). In this paper, as a continuation of the study of the Hermite-Hadamard inequality, we establish some results for k-Riemann-Liouville fractional integral by using the definition of convex functions via fractional calculus.
Below, let us recall first some basic concepts and some earlier results.
respectively, where is the classical Gamma function and I 0 .

Hermite-Hadamard's inequalities for k-fractional integrals
In [3], the k-gamma function was introduced by Diaz et al. as follows.

Definition 3
Let k and R(v) be positive. Then the k-gamma function is defined by following integral: and respectively. Here k is the k-Gamma function.

be the left and right sided k-Riemann-Liouville fractional integral of order
Multiplying both sides of (10) by μ λ k -1 , then integrating with respect to μ over [0, 1], we get where By taking μa + (1μ)b = φ in I 1 and μb + (1μ)a = ω in I 2 , we get and Substituting the values of I 1 and I 2 from (12) and (13) in (11), we get which implies that This completes the first inequality in (8). To complete the second inequality, we note that if f is convex, then, for τ ∈ [0, 1], it yields that By adding the above two inequalities, we get Multiplying by μ λ k -1 on both sides of (15), then integrating with regard to μ over [0, 1], we get We denote Putting φ = μa + (1μ)b in K 1 , and ω = μb + (1μ)a in K 2 , we obtain and Substituting the values of K 1 and K 2 from (17) and (18) in (16), we get which implies that By combining (17), and (19), we get (8).
Proof Let us consider which, we can write as Integrating I 1 by parts, we get Setting ξ = μa + (1μ)b, then after some calculation, we get Now integrating I 2 by parts to get Setting ξ = (μa + (1μ)b), after some calculation, we get Applying (22) and (23) in (21), it follows that Multiplying both sides of (24) by b-a 2 to get the required result.
Proof By using Lemma 2.1 and the definition of a convex function of |f |, we have where and We calculate K 1 to get Similarly we can calculate K 2 and get Substituting the values of K 1 and K 2 in (26) and after some calculations, we get (25).

Some more fractional inequalities for convex functions
The k-Riemann-Liouville integrals I λ ( a+b 2 ) + ,k f and I λ ( a+b 2 ) -,k f of order λ > 0 and k > 0 with a ≥ 0 are defined by respectively. Here k (λ) is the k-Gamma function.