New Hermite–Hadamard type inequalities are obtained for convex functions via generalized fractional integrals. The results presented here are generalizations of those obtained in earlier works. | Turk J Math (2016) 40: 1221 – 1230 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On Hermite–Hadamard type inequalities via generalized fractional integrals Mohamed JLELI1 , Donal O’REGAN2 , Bessem SAMET1,∗ Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia 2 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland 1 Received: • Accepted/Published Online: • Final Version: Abstract: New Hermite–Hadamard type inequalities are obtained for convex functions via generalized fractional integrals. The results presented here are generalizations of those obtained in earlier works. Key words: Hermite–Hadamard inequality, convex function, generalized fractional integral, Riemann–Liouville fractional integral, Hadamard fractional integral 1. Introduction Let I be an interval of real numbers and a, b ∈ I with a 0 , Γ is the Gamma function, and Jaα+ f and Jbα− f are the left-sided and right-sided Riemann– Liouville fractional integrals of order α > 0 . Note that for α = 1, () reduces to the classical Hermite–Hadamard inequality (). Theorem Let f : ˚ I → R be a differentiable mapping on ˚ I , a, b ∈ ˚ I with a 0 . Observe that for α = 1, () reduces to (). In this paper, we obtain generalizations of Theorems and using the generalized fractional integrals introduced recently by Katugampola in [16]. First we recall some definitions and mathematical preliminaries that will be used in this paper. Let f : [a, b] → R be a given function, where 0 0 of f is defined by Jaα+ f (x) = 1 Γ(α) ∫ x (x − τ )α−1 f (τ ) dτ, x > a, () a provided that the integral exists. The right-sided Riemann–Liouville fractional integral Jbα− of order α > 0 of f is defined by Jbα− f (x) = 1 Γ(α) ∫ b (τ − x)α−1 f (τ ) dτ, x 0 of f is defined by Jα a+ f .
