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 Improvement of Aczél’s Inequality and Popoviciu’s Inequality | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 72173 9 pages doi 2007 72173 Research Article Improvement of Aczel s Inequality and Popoviciu s Inequality Shanhe Wu Received 30 December 2006 Accepted 24 April 2007 Recommended by Laszlo I. Losonczi We generalize and sharpen Aczel s inequality and Popoviciu s inequality by means of two classical inequalities a unified improvement of Aczel s inequality and Popoviciu s inequality is given. As application an integral inequality of Aczel-Popoviciu type is established. Copyright 2007 Shanhe Wu. 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 1956 Aczel 1 proved the following result n n _ V n2 b2 - V b2 a1 2 at b1 A bi -I i 2 i 2 a1b iatb 2 i 2 1 T A rTs z 7 . fl . Í t _ 1 o lz 1 ZT 1 o TA ZA CTÍh t a TA 1 I m 1a Ol C oil Z 1a f 1a a Í XT2 _ XI2 A í ZA V fl 2 _ fl 2 A where ai bi i 1 2 . n are positive nu QDers such that a1 i 2 ai 0 or b1 i 2 bi 0. This inequality is called Aczel s inequality. It is well known that Aczel s inequality has important applications in the theory of functional equations in non-Euclidean geometry. In recent years this inequality has attracted the interest of many mathematicians and has motivated a large number of research papers involving different proofs various generalizations improvements and applications see 2-11 and references therein . We state here a brief history on improvement of Aczel s inequality. Popoviciu 12 first presented an exponential extension of Aczel s inequality as follows. 2 Journal of Inequalities and Applications Theorem . Let p 0 q 0 1 p 1 q 1 and let Uị bị i 1 2 . n be positive íí IÍ i li Z ííí np _ n np 0 Í7 t hi _ n . td n rÌ tii ll iiumeissuc t at a1 1i 2 ai an 1 1i 2 i . en n 1 p n 1 q n p p q q a - a i b - A bi a1b1 - A aibi. .
