Hermite-Hadamard type inequalities for the generalized k-fractional integral operators

We firstly give a modification of the known Hermite-Hadamard type inequalities for the generalized k-fractional integral operators of a function with respect to another function. We secondly establish several Hermite-Hadamard type inequalities for the generalized k-fractional integral operators of a function with respect to another function. The results presented here, being very general, are pointed out to be specialized to yield some known results. Relevant connections of the various results presented here with those involving relatively simple fractional integral operators are also indicated.


Introduction and preliminaries
A function f : I → R is said to be convex if the following inequality holds: where I is an interval in the real line R. Here and in the following, let C, R, R + , and N be the sets of complex numbers, real numbers, positive real numbers, and positive integers, and let N  := N ∪ {} and R +  := R + ∪ {}. One of the best-known inequalities for convex functions is the following Hermite-Hadamard inequality: If f : I ⊆ R → R (I is an interval) is a convex function and a, b ∈ I with a < b, then The Hermite-Hadamard inequality in () has attracted many mathematicians' attention who have presented a variety of generalizations, extensions, and variants, which are called Hermite-Hadamard type inequalities (see, e.g., [-] and the references cited therein).
Recently, several Hermite-Hadamard type inequalities associated with fractional integrals have been investigated. Here, we aim to establish several generalized Hermite-Hadamard type integral inequalities for the generalized k-fractional integral operators with respect to another function. The results presented here, being very general, are also pointed out to be specialized to yield some known results. Relevant connections of the various results presented here with those involving relatively simple fractional integral operators are also indicated.
To do this, we recall some definitions and known results. Let [a, b] (-∞ < a < b < ∞) be a finite interval on the real axis R. The Riemann-Liouville fractional integrals J α a+ f and J α b-f of order α ∈ C ( (α) > ) with a ≥  and b >  are defined, respectively, by Here (α) is the familiar Gamma function (see, e.g., [], Section .). For more details and properties of the fractional integral operators () and (), we refer the reader, for example, to [-] and the references therein.
be a finite or infinite interval on the real axis R. We denote by L p (a, b) ( ≤ p ≤ ∞) the set of those Lebesgue complex-valued measurable functions f on for which f p < ∞, where In particular, L  (a, b) := L(a, b).

Raina [] introduced a class of functions defined formally by
where the coefficients σ (m) ∈ R + (m ∈ N  ) form a bounded sequence. With the help of (), Raina [] and Agarwal et al.
[] defined, respectively, the following left-sided and right-sided fractional integral operators: and where λ, ρ ∈ R + , w ∈ R, and ϕ(t) is a function such that the integrals on the right sides exist. Recently, certain new and interesting inequalities involving these fractional operators have appeared in the literature (see, e.g., [-]). It is easy to verify that J σ ρ,λ,a+;w ϕ(x) and J σ ρ,λ,b-;w ϕ(x) are bounded integral operators on In fact, and Here, many useful fractional integral operators can be obtained by specializing the function F σ ρ,λ (x). For instance, the classical Riemann-Liouville fractional integrals J α a+ and J α bof order α follow easily by setting λ = α, σ () = , and w =  in () and (). Budak et al.
[] established a new identity involving the fractional integral operators () and () asserted by the following lemma.
We recall the following generalized fractional integral operators (see, e.g., be an increasing and positive function having a continuous derivative g on (a, b). The left-and right-sided generalized fractional integrals of f with respect to the function g on [a, b] of order α are defined, respectively, by and provided that the integrals exist. The integrals ()) and () are usually called fractional integrals of a function f by a function g of the order α. Choosing g(x) = x in () and We recall some properties for the k-gamma function: Using the k-gamma function, Tunç et al.
[] introduced a class of functions defined by is a bounded sequence as given in (). Tunç et al.
[] used the function () to define the left-sided and right-sided generalized k-fractional integral operators with respect to another function as follows: Let k, ρ, λ ∈ R + and w ∈ R. Also, let g : [a, b] → R be an increasing and positive function having a continuous derivative g on (a, b). Then the left-and right-sided generalized k-fractional integrals of f with respect to the function g on [a, b] are defined, respectively, by in the integral operator () gives the generalized fractional integral operator of f with respect to the function g, the generalized k-fractional integral operator of f on [a, b], the generalized Hadamard kfractional integral operator of f , and the generalized (k, s)-fractional integral operator of f on [a, b], respectively, as follows: The special cases of () and () when k =  and g(t) = t reduce to yield the generalized fractional integral operators () and (), respectively (see [, ]). Further, setting k = , g(t) = t, λ = α, σ () = , and w =  in () and () gives, respectively, the Riemann-Liouville fractional integrals () and (). The Hermite-Hadamard type inequalities in [] have been generalized by Tunç et al.
[] who used the generalized k-fractional integral operators () and (), which is recalled in the following theorem.
where F(x) is defined as in ().

Hermite-Hadamard type inequalities for fractional integral operators
We begin by recalling some notations given in [].
in such a way that I α a+,g f (x) and I α b-,g f (x) are well defined. We define the following functions: and Also, the following notations will be used throughout this paper: Taking s =  in () and (), respectively, gives The Hermite-Hadamard type inequalities for the generalized k-fractional integrals of a function with respect to another function in Theorem  can be modified as in the following theorem.
Theorem  Let k, ρ, λ ∈ R + , w ∈ R +  , and σ (m) ∈ R + (m ∈ N  ) be a bounded sequence. Also, let g : It is easy to see that Multiplying both sides of () by and integrating the resulting inequality on [, ] with respect to s, with the aid of (), (), (), (), (), and (), we obtain Similarly, multiplying both sides of () by and integrating the resulting inequality on [, ] with respect to s, with the aid of (), (), (), (), (), and (), we get From () and (), we have which proves the first inequality in ().
To prove the second inequality in (), using the convexity of f on [a, b], we obtain By adding these inequalities, we get Multiplying both sides of () by and integrating the resulting inequality on [, ] with respect to s, similar to the proof of the first inequality, we have Similarly, multiplying both sides of () by and integrating the resulting inequality on [, ] with respect to s, we obtain Adding () and (), we have which proves the second inequality in (). Hence this completes the proof.
Setting k =  in Theorem , we get a little simpler inequalities asserted by the following corollary.

[a, b] → R be an increasing and positive function on [a, b] having a continuous derivative g (x) on (a, b). If f is a convex function on [a, b], then the following Hermite-Hadamard type inequalities for the generalized fractional integrals of f with respect to the function g on [a, b] in () and () with
where F(x) is defined as in ().
Further, choosing λ = α, σ () =  and w =  in Corollary , we get simpler inequalities in the following corollary, which are a modification of the Hermite-Hadamard inequalities given in []. (a, b). If f is a convex function on [a, b], then the following Hermite-Hadamard type inequalities for the generalized fractional integrals of f with respect to the function g on [a, b] in () and () hold:

Corollary  Let α ∈ R + and g : [a, b] → R be an increasing and positive function on [a, b] having a continuous derivative g (x) on
where F(x) is defined as in ().
It is remarked in passing that choosing g(t) = t in Corollary  yields the same result as in [], Corollary .

Main results
We begin by presenting an integral formula involving the functions () and (), which is asserted by the following lemma.
Proof Using () and changing the variable Integrating () by parts, we have Similarly, using () and changing the variable and integrating the resulting identity by parts, we have Using () to add () and (), we obtain and applying ()-() to (), we obtain the desired identity ().
Setting k =  in Lemma , we obtain an identity asserted by the following corollary.
where the notations are given as above and Theorem  Let k, ρ, λ ∈ R + , w ∈ R +  , and σ (m) ∈ R + (m ∈ N  ) be a bounded sequence. Also, let g : [a, b] → R be an increasing and positive function on [a, b] having a continuous derivative g (x) on (a, b). Further, let f : [a, b] → R be a differentiable mapping on (a, b) (a < b) such that |f | q is convex for q ≥ . Then where the notations are given above: and Proof Using convexity of |f | q and the power-mean inequality in Lemma , we have where We, therefore, have This completes the proof.