tailieunhanh - New inequalities for fractional integrals and their applications

In this paper, we establish some Hermite–Hadamard-type, Bullen-type, and Simpson-type inequalities for fractional integrals. Some applications for the beta function are also given. | Turk J Math (2016) 40: 471 – 486 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article New inequalities for fractional integrals and their applications Shiow-Ru HWANG1 , Kuei-Lin TSENG2,∗, Kai-Chen HSU2 China University of Science and Technology, Nankang, Taipei, Taiwan 2 Department of Applied Mathematics, Aletheia University, Tamsui, New Taipei City, Taiwan 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we establish some Hermite–Hadamard-type, Bullen-type, and Simpson-type inequalities for fractional integrals. Some applications for the beta function are also given. Key words: Hermite–Hadamard inequality, Bullen inequality, Simpson inequality, fractional integral, convex function 1. Introduction Throughout this paper, let a 0 with a ≥ 0 are defined by Jaα+ f 1 (x) = Γ (α) and Jbα− f (x) = 1 Γ (α) ∫ x α−1 (x − t) f (t) dt (x > a) f (t) dt (x 0. Theorem E Under the assumptions of Theorem B, then we have f (a) + f (b) Γ (α + 1) α α − α [Ja+ f (b) + Jb− f (a)] 2 2 (b − a) ≤ for α > 0. 472 2α − 1 (b − a) (|f ′ (a)| + |f ′ (b)|) 2α+1 (α + 1) () HWANG et al./Turk J Math Zhu et al. [16] established the following fractional integral inequality with the first inequality of (): Theorem F Under the assumptions of Theorem B, then we have ) ( Γ (α + 1) α a + b α 2 (b − a)α [Ja+ f (b) + Jb− f (a)] − f 2 ( ) (b − a) 1 ≤ α + 3 − α−1 (|f ′ (a)| + |f ′ (b)|) 4 (α + 1) 2 () for α > 0. Remark 1. The assumption f : [a, b] → R is positive with 0 ≤ a 0. Proof Define { h1 (x) = [ ) α α α (b − x) − (x − a) − (b − a) , x ∈ [a, a+b 2 ] . α α α (b − x) − (x − a) + (b − a) , x ∈ a+b 2 ,b 473 HWANG et al./Turk J Math Using integration by parts, we have the following identities: 1 α 2 (b − a) = α α 2 (b − a) = αΓ (α) α 2 (b − a) = Γ (α + 1) α 2 (b − a) ∫ b h1 (x) f ′ (x)