TAILIEUCHUNG - 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) (

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