Generalized fractional Hadamard type inequalities for Qs–class functions of the second kind

H ermite–Hadamard (HH) inequalities and other related inequalities for convex functions have been extensively studied by different researchers, see [1,2] and their references. A much broader class of functions known as Q–class functions was proposed by Godunova and Levin [3,4]. This class of functions is very important because it contains all nonnegative monotone and nonnegative convex functions. Motivated by the fact that this class of functions is much bigger and broader than the class of convex functions (which many authors have given different HH and HH type inequalities on); we, therefore, present, extend and generalize the HH inequalities on this broader class of functions for fractional integrals.


Introduction and Preliminaries
H ermite-Hadamard (HH) inequalities and other related inequalities for convex functions have been extensively studied by different researchers, see [1,2] and their references. A much broader class of functions known as Q-class functions was proposed by Godunova and Levin [3,4]. This class of functions is very important because it contains all nonnegative monotone and nonnegative convex functions. Motivated by the fact that this class of functions is much bigger and broader than the class of convex functions (which many authors have given different HH and HH type inequalities on); we, therefore, present, extend and generalize the HH inequalities on this broader class of functions for fractional integrals. Definition 1 ([5-9]). A nonnegative function f : I → R is said to be a Q-class function, if for all x, y ∈ I, and λ ∈ (0, 1), Definition 2 ([5,6]). Let D be a subset of R with at least two elements. A function f : for all x, y, z ∈ D. [3] also showed that the class of Schur functions and the Q-class functions are equivalent. That is, (1) and (2) concide.

Definition 3 ([5,6]).
A nonnegative function f : I → R is said to be a P-function, if ∀ x, y ∈ I, λ ∈ [0, 1], It is also known that P(I) ⊂ Q(I), and contains all nonnegative monotone, convex and quasi-convex functions: nonnegative functions satisfying The following Hamadard type inequalities have already been proved: [4][5][6]). Let f ∈ Q(I), a, b ∈ I, with a < b and f ∈ L 1 [a, b]. Then Next, we state some generalizations of the Q-class function known as the s-Godunova-Levin functions (Q s -class functions): . A nonnegative function f : I → R is said to be a Q s -class functions of first kind if for s ∈ (0, 1], ∀ x, y ∈ I, λ ∈ (0, 1).

Definition 5.
A function f : I → R is said to be a Q s -class functions of second kind if for s ∈ [0, 1], ∀x, y ∈ I, λ ∈ (0, 1).

Remark 3.
Observe that s = 0 in (5) gives the definition of P-class function in (3), and s = 1 gives the definition of Q-class function in (1).
The paper is organized as follows. Section 2 contains the main results of the paper. In Section 3, we give a concise summary of the paper.

Main Results
Our aim is to extend and generalize the result of Theorem 1 to s-Godunova-Levin functions of second kind given in (5):

Some Auxilliary Results
Integrate both sides of (7) over t ∈ [0, 1] to obtain Let Also, when t = 0, z = b and when t = 1, z = a. Thus, Similarly, for I 2 , we let u = (1 − t)a + tb), u − a = (b − a)t and du = (b − a)dt; when t = 0, u = a and when t = 1, u = b. Therefore, Combining I 1 and I 2 , Inequality (8) , and the first part of the result follows. For the second part of the proof, we multiply (5) Similarly, Now, add (9) and (10), and integrate over λ ∈ [0, 1]: Evaluating the integrals as before, we have One can apply the Integral Chebyshev inequality, to get alternative inequalities of (6) of Theorem 2: Suppose that the hypotheses of Theorem 2 hold. Then Proof. Recall from the proof of Theorem 2, that Now, applying the Integral Chebyshev inequality on the integrals Thus, So, we obtain that and the first inequality follows. Next, we write Inequality (5) for a, b as follows: and integrate over λ ∈ [0, 1] to get

Conclusion
The results focus on new generalized fractional Hadamard type inequalities for s-Godunova-Levin functions of the second kind. The obtained results generalize and extend already existing results.