Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : An extension of Jensen’s discrete inequality to half convex functions | Cirtoaje and Baiesu Journal of Inequalities and Applications 2011 2011 101 http www.journalofinequalitiesandapplications.eom content 2011 1 101 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access An extension of Jensen s discrete inequality to half convex functions Vasile Cirtoaje and Alina Baiesu Correspondence vcirtoaje@upg-ploiesti.ro Department of Automatic Control and Computers University of Ploiesti 100680 Ploiesti Romania Springer Abstract We extend the right and left convex function theorems to weighted Jensen s type inequalities and then combine the new theorems in a single one applicable to a half convex function f u defined on a real interval D and convex for u s or u s where s e D. The obtained results are applied for proving some open relevant inequalities. Keywords weighted Jensen s discrete inequality right convex function left convex function half convex function 1 Introduction The right convex function theorem RCF-Theorem has the following statement see 1-3 . RCF-Theorem. Let f u be a function defined on a real interval D and convex for u s e D. The inequality f X1 f X2 f xn nf X1 Xn 1 n holds for all X1 X2 . Xn e satisfying X1 X2 . Xn ns if and only if f x n - 1 f y nf s 2 for all X y e D which satisfy X s y and X n - 1 y ns. Replacing f u by f -u s by -s X by -X y by -y and each xi by -xi for i 1 2 . n from RCF-Theorem we get the left convex function theorem LCF-Theorem . LCF-Theorem. Let f u be a function defined on a real interval D and convex for u s e D. The inequality X1 X2 Xn f X1 f X2 f Xn nf -------- 3 n holds for all X1 X2 . Xn e satisfying X1 X2 . Xn ns if and only if f x n - 1 f y nf s 4 for all X y e D which satisfy X s y and X n - 1 y ns. Notice that from RCF- and LCF-Theorems we get the following theorem which we have called the half convex function theorem HCF-Theorem . 2011 Cirtoaje and Baiesu licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons .
