New general integral inequalities for Lipschitzian functions via Hadamard fractional integrals

In this paper, the author obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive reals numbers are also given.


Introduction
Let real function f be defined on some nonempty interval I of real line R. The function f is said to be convex on I if inequality holds for all x, y ∈ I and t ∈ [0, 1] . Following inequalities are well known in the literature as Hermite-Hadamard inequality, Ostrowski inequality and Simpson inequality respectively: Theorem 1. Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following double inequality holds Theorem 2. Let f : I ⊆ R → R be a mapping differentiable in I • , the interior of I, and let a, b ∈ I • with a < b. If |f ′ (x)| ≤ M, x ∈ [a, b] , then we the following inequality holds for all x ∈ [a, b] . f (4) (x) < ∞. Then the following inequality holds: respectively, where Γ(α) is the Gamma function defined by Γ(α) = ∞ 0 e −t t α−1 dt (see [9]).
In [8], Iscan established Hermite-Hadamard's inequalities for GA-convex functions in Hadamard fractional integral forms as follows.
, then the following inequalities for fractional integrals hold: with α > 0.
In the inequality (1.3), if we take α = 1, then we have the following inequality Morever in [8], Iscan obtained a generalization of Hadamard, Ostrowski and Simpson type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals as related the inequality (1.3).
In this paper, the author obtains new general inequalities for Lipschitzian functions via Hadamard fractional integrals as related the inequality (1.3).

Main Results
Let f : I ⊆ (0, ∞) → R be a differentiable function on I • , the interior of I, throughout this section we will take Theorem 5. Let f : I ⊆ (0, ∞) → R be a M -Lipschitzian function on I and a, b ∈ I with a < b. then for all x ∈ [a, b] , λ ∈ [0, 1] and α > 0 we have the following inequality for Hadamard fractional integrals Proof. Using the hypothesis of f , we have the following inequality Corollary 1. In Theorem 5, If we take λ = 0, then we get In this inequality, Corollary 2. In Theorem 5, If we take λ = 1, then we get In this inequality, if we take x = √ ab, then Specially if we take α = 1 in this inequality, then we have Corollary 3. In Theorem 5, (1) If we take x = √ ab and λ = 1/3, then Specially if we take α = 1 in this inequality, then we have
Theorem 6. Let x, y, α, λ, C, C α,λ and the function f be defined as above. Then we have the following inequality for hadamard fractional integrals Proof. Using the hypothesis of f , we have the following inequality Now using simple calculations, we obtain the following identities Using the inequality (2.9) and the above identities Under the assumptions of Theorem 6, we have the following corollaries and remarks: Corollary 5. In Theorem 6, if we take α = 1, then the inequality (2.8) reduces the following inequality: Corollary 6. In Theorem 6, let δ ∈ 1 2 , 1 , x = a δ b 1−δ and y = a 1−δ b δ . Then, we have the inequality Corollary 7. In Theorem 6, if we take x = y = C, then we have the inequality Remark 1. In the inequality (2.11), if we choose λ = 1 2 , then we get the inequality (2.1).
Corollary 8. In the inequality (2.10), if we take δ = 1, then we have the following weighted Hadamard-type inequalities for Lipschitzian functions via Hadamard fractional integrals Remark 2. In the inequality (2.12), if we choose λ = 1 2 , then we get the inequality (2.3).

Application to Special Means
Let us recall the following special means of two positive number a, b with b > a : 10İMDATİŞCAN (1) The arithmetic mean (2) The geometric mean (3) The harmonic mean (4) The logarithmic mean (5) The identric mean To prove the results of this section, we need the following lemma: Proposition 1. For b > a > 0, λ ∈ [0, 1] and n ≥ 1, we have.