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: On improvements of the Rozanova’s inequality | Zhao and Cheung Journal of Inequalities and Applications 2011 2011 33 http content 2011 1 33 3 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access On improvements of the Rozanova s inequality Chang-Jian Zhao 1 and Wing-Sum Cheung2 Correspondence chjzhao@163. com department of Mathematics China Jiliang University Hangzhou 310018 China Full list of author information is available at the end of the article Abstract In the present paper we establish some new Rozanova s type integral inequalities involving higher-order partial derivatives. The results in special cases yield some of the interrelated results on Rozanova s inequality and provide new estimates on inequalities of this type. MS 2000 Subject Classification 26D15. Keywords Opial s inequality Holder s inequality Rozanova s inequality 1 Introduction In the year 1960 Opial 1 established the following integral inequality Theorem A Supposef e C1 0 h satisfies fữ fh 0 andfx 0for all x e 0 h . Then ch h Í h f x f x dx f x 2dx. 1 1 0 4 0 The first Opial s type inequality was established by Willett 2 as follows Theorem B Let x t be absolutely continuous in 0 a and x 0 0. Then a a I a J x t X t dt 2 J X t 2dt. A non-trivial generalization of Theorem B was established by Hua 3 as follows Theorem C Let x t be absolutely continuous in 0 a and x 0 0. Futher let l be a positive integer. Then ca al ca J x t x t dt J x t l 1dt. A sharper inequality was established by Godunova 4 as follows Theorem D Let f t be convex and increasing functions on 0 with f 0 0. Further let x t be absolutely continuous on 0 t and x a 0. Then following inequality holds f x t x t dt f a a x t dộ . Rozanova 5 proved an extension of inequality is embodied in the following Theorem F Let ft and g t be convex and increasing functions on 0 withf0 0 and let p t 0 p t 0 t e a a with p a 0. Further let x t be absolutely SpringerOpen0 2011 Zhao and Cheung licensee .
