Abstract
Abstract In the present note, we have given a new integral identity via Conformable fractional integrals and some further properties. We have proved some integral inequalities for different kinds of convexity via Conformable fractional integrals. We have also showed that special cases of our findings gave some new inequalities involving Riemann-Liouville fractional integrals.
Highlights
We will recall some preliminaries concepts to refresh our memories: Definition 1. [See [5]] A mapping f : I → [0, ∞) is said to be log−convex or multiplicatively convex if log f is convex or equivalently for all θ, θ ∈ I and ς ∈ [0, 1], one has the inequality:f (ς θ + (1 − ς )θ ) ≤ [ f (θ )]ς [ f (θ )]1−ς (1.1)We note that a log −convex function satisfy the condition of convexity, but the converse may not necessarily be true.Definition 2. [See [6]] Let s ∈
In [7], s−convexity introduced by Breckner as a generalization of convex functions
The results which are obtained by using the conformable fractional integrals have a wide range of validity. (Let us consider the function f defined as f : R+ → R, f = x2ex which is convex.)
Summary
We will recall some preliminaries concepts to refresh our memories: Definition 1. [See [5]] A mapping f : I → [0, ∞) is said to be log−convex or multiplicatively convex if log f is convex or equivalently for all θ , θ ∈ I and ς ∈ [0, 1], one has the inequality:f (ς θ + (1 − ς )θ ) ≤ [ f (θ )]ς [ f (θ )]1−ς (1.1)We note that a log −convex function satisfy the condition of convexity, but the converse may not necessarily be true.Definition 2. [See [6]] Let s ∈ (See [5]) A function f : [a, b] → R is said quasi-convex on [a, b] if f (λ x + (1 − λ )y) ≤ max { f (x), f (y)} , (QC) Let us recall the definition of Riemann-Liouville fractional integrals: Definition 4. The Riemann-Liouville integrals Jaα+ f and Jbα− f of order α > 0 are defined by
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