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Báo cáo hóa học: " Research Article On an Inequality of H. G. Hardy"

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 On an Inequality of H. G. Hardy | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 264347 23 pages doi 10.1155 2010 264347 Research Article On an Inequality of H. G. Hardy Sajid Iqbal 1 Kristina Krulic 2 and Josip Pecaric1 2 1 Abdus Salam School of Mathematical Sciences GC University Lahore 54600 Pakistan 2 Faculty of Textile Technology University of Zagreb Prilaz baruna Filipovica 28a 10000 Zagreb Croatia Correspondence should be addressed to Sajid Iqbal sajiduos2000@yahoo.com Received 18 June 2010 Accepted 16 October 2010 Academic Editor Q. Lan Copyright 2010 Sajid Iqbal et al. 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. We state prove and discuss new general inequality for convex and increasing functions. As a special case of that general result we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals. 1. Introduction First let us recall some facts about fractional derivatives needed in the sequel for more details see for example 1 2 . Let 0 a b TO. By Cm a b we denote the space of all functions on a b which have continuous derivatives up to order n and AC a b is the space of all absolutely continuous functions on a b . By ACm a b we denote the space of all functions g e Cm a b with g m-1 e AC a b . For any a e R we denote by a the integral part of a the integer k satisfying k a k 1 and a is the ceiling of a min n e N n a . By L1 a b we denote the space of all functions