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: Research Article Steffensen’s Integral Inequality on Time Scales | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 46524 10 pages doi 2007 46524 Research Article Steffensen s Integral Inequality on Time Scales Umut Mutlu Ozkan and Huseyin Yildirim Received 9 May 2007 Revised 13 June 2007 Accepted 29 June 2007 Recommended by Martin J. Bohner We establish generalizations of Steffensen s integral inequality on time scales via the diamond-a dynamic integral which is defined as a linear combination of the delta and nabla integrals. Copyright 2007 U. M. Ozkan and H. Yildirim. 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 Steffensen 1 stated that if f and g are integrable functions on a b with f nonincreasing and 0 g 1 then fb f t dt fb f t g t dt fa Ảf t dt -A a a where A ag t dt. This inequality is usually called Steffensen s inequality in the literature. A comprehensive survey on Steffensen s inequality can be found in 2 . Recently Anderson 3 has given the time scale version of Steffensen s integral inequality using nabla integral as follows let a b e T and let f g a b T R be nabla integrable functions with f of one sign and decreasing and 0 g 1 on a b T. Assume Ể Y e a b T such that c b b t g t Vt Y - a if f 0 t e a b T a b Y a g t Vt b t if f 0t e a b T. a 2 Journal of Inequalities and Applications Then b r b r Y f t Vt f t g t Vt f t Vt. e a a In the theorem above which can be found in 3 as Theorem we could replace the nabla integrals with delta integrals under the same hypotheses and get a completely analogous result. Wu 4 has given some generalizations of Steffensen s integral inequality which can be written as the following inequality let f g and h be integrable functions defined on a b with f nonincreasing. Also let 0 g t h t t e a b . Then fb f t h t dt fb f t g t dt fa Af t h
