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Báo cáo hóa học: " Research Article Fractional Quantum Integral Inequalities ¨ ˘¨ ¨ Hasan Ogunmez and Umut Mutlu Ozka"

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 Fractional Quantum Integral Inequalities ¨ ˘¨ ¨ Hasan Ogunmez and Umut Mutlu Ozka | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 787939 7 pages doi 10.1155 2011 787939 Research Article Fractional Quantum Integral Inequalities Hasan Ogunmez and Umut Mutlu Ozkan Department of Mathematics Faculty of Science and Arts Kocatepe University 03200 Afyonkarahisar Turkey Correspondence should be addressed to Umut Mutlu Ozkan umukozkan@aku.edu.tr Received 10 November 2010 Revised 19 January 2011 Accepted 16 February 2011 Academic Editor J. Szabados Copyright 2011 H. Ogunmez and U. M. Ozkan. 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. The aim of the present paper is to establish some fractional q-integral inequalities on the specific time scale Tt0 t t t0qn n a nonnegative integer u 0 where t0 e JR and 0 q 1. 1. Introduction The study of fractional q-calculus in 1 serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale TTt0 t t t0qn n a nonnegative integer u 0 where t0 e R and 0 q 1. Belarbi and Dahmani 2 gave the following integral inequality using the Riemann-Liouville fractional integral if f and g are two synchronous functions on 0 to then Jafg t Jaf J t 1.1 for all t 0 a 0. Moreover the authors 2 proved a generalized form of 1.1 namely that if f and g are two synchronous functions on 0 to then ta . . tp . . ft Ja fg t Jaf J t Jpf u t 1.2 1 a T1J r p 11 for all t 0 a 0 and p 0. 2 Journal of Inequalities and Applications Furthermore the authors 2 pointed out that if fi i 12 n are n positive increasing functions on 0 to then Ja rtfi 9 ĩaf 1 1-n nw i 1 1 1 1.3 for any t 0 a 0. In this paper we have obtained fractional q-integral inequalities which are quantum versions of inequalities 1.1 1.2 and 1.3 on the specific time scale TTt0 t t t0qn n a nonnegative integer u 0 where to 6 R and 0