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Báo cáo hóa học: " New inequalities of hermite-hadamard type for convex functions with applications"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : New inequalities of hermite-hadamard type for convex functions with applications | Kavurmaci et al. Journal of Inequalities and Applications 2011 2011 86 http www.journalofinequalitiesandapplications.eom content 2011 1 86 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access New inequalities of hermite-hadamard type for convex functions with applications Havva Kavurmaci Merve Avci and M Emin Ozdemir Correspondence hkavurmaci@atauni.edu.tr Aataturk University K.K. Education Faculty Department of Mathematics 25240 Campus Erzurum Turkey Abstract In this paper some new inequalities of the Hermite-Hadamard type for functions whose modulus of the derivatives are convex and applications for special means are given. Finally some error estimates for the trapezoidal formula are obtained. 2000 Mathematics Subject Classiffication. 26A51 26D10 26D15. Keywords Convex function Hermite-Hadamard inequality Holder inequality Powermean inequality Special means Trapezoidal formula 1. Introduction A function f I R is said to be convex function on I if the inequality f ax 1 a y af x 1 a f y holds for all x y e I and a e 0 1 . One of the most famous inequality for convex functions is so called Hermite-Hada-mard s inequality as follows Let f I R R be a convex function defined on the interval I of real numbers and a b e I with a b. Then f ĩff. _L. f f x dx MM J. 1.1 2 b a a 2 In 1 the following theorem which was obtained by Dragomir and Agarwal contains the Hermite-Hadamard type integral inequality. Theorem 1. Let f r R R be a differentiable mapping on r a b e I with a b. If If is convex on a b then the following inequality holds f a f b 1 b-a mi If b l t fu ----------------s---------- 1 2 In 2 Kirmaci Bakula Ozdemir and Pecaric proved the following theorem. Theorem 2. Let f I R I c R be a differentiable function on r such thatf e L a b where a b e I a b. If lf lq is concave on a b for some q 1 then Springer 2011 Kavurmaci et al licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution