J. Math. Computer Sci., 22 (2021), 363–380 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs Fractional Opial dynamic inequalities A. G. Sayed a , S. H. Saker b,∗ , A. M. Ahmed a,c a Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt. b Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. c Department of Mathematics, College of Science, Jouf University, Sakaka (2014), Kingdom of Saudi Arabia. Abstract In this paper, we will prove some new fractional dynamic inequalities on time scales of Opial’s type. The results will be proved by employing the chain rule and H¨ older’s inequality on fractional time scales. As a special case of our results, when α = 1, we will obtain several well-known dynamic Opial inequalities on time scales. Keywords: Opial’s inequality, H ¨ older’s inequality, time scales, conformable fractinonal calculus. 2020 MSC: 26A15, 26D10, 26D15, 39A13, 34A40, 34N05. c 2021 All rights reserved. 1. Introduction In 2001, Bohner and KaymakC ¸ alan [9] proved some dynamic inequalities of Opial type on time scales. One of the inequalities proved in [9] is given by  a 0 |y(t)+ y σ (t)|   y Δ (t)   ∆t  a  a 0   y Δ (t)   2 ∆t, where y :[0, a] ∩ T → R is delta differentiable with y(0)= 0. Also Bohner and KaymakC ¸ alan in [9] proved that if p and q are positive functions on [0, b] T ,  b 0 (∆t/p(t)) < ∞, q is non-increasing, and y :[0, b] ∩ T → R is delta differentiable with y(0)= 0, then  b 0 q σ (t)   (y(t)+ y σ (t))y Δ (t)   ∆t   b 0 ∆t p(t)  b 0 p(t)q(t)   y Δ (t)   2 ∆t. (1.1) Karpuz et al. [17] replaced q σ (t) with q(t) and proved an inequality similar to (1.1) of the form  b a q(t)   (y(t)+ y σ (t))y Δ (t)   ∆t  K q (a, b)  b a   y Δ (t)   2 ∆t, ∗ Corresponding author Email addresses: a.g_sayed@yahoo.com (A. G. Sayed), shsaker@mans.edu.eg (S. H. Saker), ahmedelkb@yahoo.com & amaahmed@ju.edu.sa (A. M. Ahmed) doi: 10.22436/jmcs.022.04.05 Received: 2020-06-22 Revised: 2020-07-22 Accepted: 2020-07-24