Some new integral inequalities for n-times differentiable convex and concave functions

Abstract

In this work, by using an integral identity together with both the Hölder and the Power-mean integral inequalities we establish several new inequalities for n-times differentiable convex and concave mappings.

Some new integral inequalities for functions whose nth derivatives in absolute value are (?, m)-convex functions

“…(ii) In case of α = m = 1, the results are obtained in this paper reduce to the results obtained for convex functions in [16].…”

“…Recently, in the literature there are so many papers about n-times differentiable functions on several kinds of convexities. In references [1,3,4,6,16,19,20], readers can find some results about this issue. Many papers have been written by a number of mathematicians concerning inequalities for different classes of (α, m)-convex functions see for instance the recent papers [1,2,10,11,13,15] and the references within these papers.…”

New Inequalities for Stronglyr-Convex Functions

“…Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences. Many articles have been written by a number of mathematicians on convex functions and inequalities for their different classes, using, for example, the last articles [1][2][3][4][5][6] and the references in these papers.…”

Some New Integral Inequalities for <i>n</i>-times Differentiable <i>s</i>- Convex functions in the first sense

“…We will use the following Lemma [20] for we obtain the main results: Lemma 1.1: Let : ⊆ ℝ → ℝ be -times differentiable mapping on ° for ∈ ℕ and ( ) ∈ [ , ] , where , ∈° with < , we have the identity…”