On Feng Qi-type Integral Inequalities for Conformable Fractional Integrals

In this paper, we establish the generalized Qi-type inequality involving conformable fractional integrals. The results presented here would provide extensions of those given in earlier works. 1. Introduction In the last few decades, much signi…cant development of integral inequalities had been established. Integral inequalities have been frequently employed in the theory of applied sciences, di¤erential equations, and functional analysis. In the last two decades, they have been the focus of attention in [4]-[7]. Recently, especially Qi inequality, one of the integral inequalities, has been studied by many authors. Recall the famous integral inequality of Feng Qi type: Theorem 1. (Proposition 1.1, [7]) Let f(x) be di¤erentiable on (a; b) and f(a) = 0. If f 0 (x) 1 for x 2 (a; b), then (1.1) 0@ b Z a [f (t)] 3 dt 1A 0@ b Z a f (t) dt 1A2 : If 0 f 0 (x) 1, then the inequality (1.1) reverses. Theorem 2. (Proposition 1.3, [7]) Let n be a positive integer. Suppose f(x) has continuous derivative of the n th order on the interval [a; b] such that f (i) (a) 0, where 0 i n 1, f (n) (x) n!; then (1.2) 0@ b Z a [f (t)] n+2 dt 1A 0@ b Z a f (t) dt 1An+1 : In [6], Ngô et al. gave the following inequality which is one of the open problem’s solution. Theorem 3. If f is a nonnegative, continuous function on [0; 1] satisfying, (1.3) 1 Z x f (t) dt 1 x 2 2 ; Key words and phrases. Integral Inequalities, Special Functions, Fractional Calculus, Conformable Fractional Integral. 2010 Mathematics Subject Classi…cation. 26A33, 26D10, 26D15, 41A55. 1 Department of Mathematics, Faculty of Science and Arts, University of KahramanmaraŞ Sütçü · Imam, 46100, KahramanmaraŞ, Turkey E-mail address : [email protected] Department of Mathematics, Faculty of Science, University of Cumhuriyet, 58140, Sivas, Turkey E-mail address : [email protected] Department of Mathematics, Faculty of Science and Arts, University of KahramanmaraŞ Sütçü · Imam, 46100, KahramanmaraŞ, Turkey E-mail address : [email protected] Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 September 2016 doi:10.20944/preprints201609.0105.v1