tailieunhanh - Báo cáo hóa học: " Research Article Generalized Lazarevic’s Inequality and Its Applications—Part II"

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 Generalized Lazarevic’s Inequality and Its Applications—Part II | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 379142 4 pages doi 2009 379142 Research Article Generalized Lazarevic s Inequality and Its Applications Part II Ling Zhu Department of Mathematics Zhejiang Gongshang University Hangzhou Zhejiang 310018 China Correspondence should be addressed to Ling Zhu zhuling0571@ Received 21 July 2009 Accepted 30 November 2009 Recommended by Andrea Laforgia A generalized Lazarevic s inequality is established. The applications of this generalized Lazarevic s inequality give some new lower bounds for logarithmic mean. Copyright 2009 Ling Zhu. 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 Lazarevic 1 or see Mitrinovic 2 gives us the following result. Theorem . Let x Ỷ 0. Then sinh x V 2------ cosh x x holds if and only if q 3. Recently the author of this paper gives a new proof of the inequality in 3 and extends the inequality to the following result in 4 . Theorem . Let p 0 and x e 0 x . Then sinh x q sinh x p h x x 2 2-p p cosh x 2 x 2 holds if and only if q p 1. sinh x x 2 Journal of Inequalities and Applications Moreover the inequality can be extended as follows. Theorem . Let p 1 or p 8 15 and x e 0 x . Then sinh x q x p 1 - p cosh x holds if and only if q 3 1 - p . 2. Three Lemmas Lemma see 5-8 . Let f g a b R be two continuous functions whichare differentiable on a b . Further let g 0 on a b . If f g is increasing or decreasing on a b then the functions f x - f b g x - g b and f x - f a g x - g a are also increasing or decreasing on a b . Lemma see 9-11 . Let an and bn n 0 1 2 . be real numbers and let the power series A x y f 0 anxn and B x y 0 bnxn be convergent for x R. If bn 0 for n 0 1 2 . and if an bn is strictly increasing or decreasing for