TAILIEUCHUNG - On Hermite–Hadamard type inequalities via generalized fractional integrals

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 .

• Toán học
• Hermite–Hadamard inequality
• Convex function
• Generalized fractional integral
• Riemann–Liouville fractional integral
• Hadamard fractional integral
• Hermite Hadamard inequality
• Bullen inequality
• Simpson inequality
• Fractional integral