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 some Opial-type inequalities | Zhao and Cheung Journal of Inequalities and Applications 2011 2011 7 http content 2011 1 7 3 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access On some Opial-type inequalities Chang-Jian Zhao1 and Wing-Sum Cheung2 Correspondence chjzhao@163. com department of Mathematics China Jiliang University Hangzhou 310018 PR China Full list of author information is available at the end of the article Abstract In the present paper we establish some new Opial-type inequalities involving higher-order partial derivatives. Our results in special cases yield some of the recent results on Opial s inequality and also provide new estimates on inequalities of this type. MR 2000 Subject Classification 26D15 Keywords Opial s inequality Opial-type integral inequalities H o lder s inequality 1 Introduction In the year 1960 Opial 1 established the following integral inequality Theorem . Suppose f e C1 0 h satisfies fữ fih 0 andf x 0 for all x e 0 h . Then the integral inequality holds r 0 f x f x dx h í f x Ýdx 4 0 where this constant 4is best possible. Opial s inequality and its generalizations extensions and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations 2-6 . The inequality has received considerable attention and a large number of papers dealing with new proofs extensions generalizations variants and discrete analogues of Opial s inequality have appeared in the literature 7-22 . For an extensive survey on these inequalities see 2 6 . For Opial-type integral inequalities involving high-order partial derivatives see 23-27 . The main purpose of the present paper is to establish some new Opial-type inequalities involving higher-order partial derivatives by an extension of Das s idea 28 . Our results in special cases yield some of the recent results
