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 Two Inequalities for r φr and Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 471527 6 pages doi 2008 471527 Research Article Two Inequalities for rộr and Applications Mingjin Wang1 2 1 Department of Mathematics East China Normal University Shanghai 200062 China 2 Department of Information Science Jiangsu Polytechnic University Jiangsu Province 213164 Changzhou China Correspondence should be addressed to Mingjin Wang wang197913@ Received 26 August 2007 Accepted 13 November 2007 Recommended by Nikolaos S. Papageorgiou We use the q-binomial formula to establish two inequalities for the basic hypergeometric series rộr. As applications of the inequalities we discuss the convergence of q-series. Copyright 2008 Mingjin Wang. 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 and main results q-series which is also called basic hypergeometric series plays a very important role in many fields such as affine root systems Lie algebras and groups number theory orthogonal polynomials physics and so on. Inequality technique is one of the useful tools in the study of special functions. There are many papers about it 1-6 . In 1 the authors gave some inequalities for hypergeometric functions. In this paper we derive two inequalities for the basic hypergeometric series rộr which can be used to study the convergence of q-series. The main results of this paper are the following two inequalities. Theorem . Suppose ai bi and z are any real numbers such that bi 1 with i 1 2 . r. Then r pr a1 a2 . ar b1 b2 . br q z r - - z q J i 1 - hl q lbi q Theorem . Suppose ai bi and z are any real numbers such that z 0 and ai 1 bj- 1 with i 1 2 . r. Then r pr a1 a2 . ar b1 b2 . br q z r Ì z q J l i 1 lai q - bi L 2 Journal of Inequalities and Applications Before the proof of the .
