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Báo cáo hóa học: "ˇ ˇ VARIANTS OF CEBYSEV’S INEQUALITY WITH APPLICATIONS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ˇ ˇ VARIANTS OF CEBYSEV’S INEQUALITY WITH APPLICATIONS | VARIANTS OF CEBYSEV S INEQUALITY WITH APPLICATIONS M. KLARICIC BAKULA A. MATKOVIC AND J. PECARIC Received 19 December 2005 Accepted 2 April 2006 Several variants of Cebysev s inequality for two monotonic n-tuples and also k 3 nonnegative n-tuples monotonic in the same direction are presented. Immediately after that their refinements of Ostrowski s type are given. Obtained results are used to prove generalizations of discrete Milne s inequality and its converse in which weights satisfy conditions as in the Jensen-Steffensen inequality. Copyright 2006 M. Klaricic Bakula et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In 2003 Mercer gave the following interesting variant of the discrete Jensen s inequality see e.g. 8 page 43 for convex functions. Theorem 1.1 4 Theorem 1 . If f is a convex function on an interval containing n-tuple x x1 . Xn such that 0 X1 X2 Xn and w w1 . wn is positive n-tuple with n 1 Wi 1 then f X1 Xn n n - WiXij f X1 f Xn - Wif Xi . 1.1 Two years later his result was generalized as it is stated below. Theorem 1.2 1 Theorem 2 . Let a b be an interval in R a b. Let x x1 . Xn be a monotonic n-tuple in a b n and let w w1 . wn be a real n-tuple such that 0 Wk Wn k 1 . n - 1 Wn 0 1.2 Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 39692 Pages 1-13 DOI 10.1155 JIA 2006 39692 2 Variants of Cebysev s inequality with applications where Wk S 1 wi k 1 n . If function f a b R is convex then 1 1 n . a b - W Ẹ wixy f a f b Wn wif x 1.3 As we can see here the condition wi 0 i 1 n is relaxed on the conditions 1.2 as in the well-known Jensen-Steffensen inequality for sums see e.g 8 page 57 . Remark 1.3. It can be easily proved that for a real n-tuple w which satisfies 1.2 and for any monotonic n-tuple x e a b n the inequalities 1 n a - W