tailieunhanh - Báo cáo hóa học: " Research Article Generalization of an Inequality for Integral Transforms with Kernel and Related Results"

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 Generalization of an Inequality for Integral Transforms with Kernel and Related Results | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 948430 17 pages doi 2010 948430 Research Article Generalization of an Inequality for Integral Transforms with Kernel and Related Results Sajid Iqbal 1 J. Pecaric 1 2 and Yong Zhou3 1 Abdus Salam School of Mathematical Sciences GC University Lahore 54000 Pakistan 2 Faculty of Textile Technology University of Zagreb 10000 Zagreb Croatia 3 School of Mathematics and Computational Science Xiangtan University Hunan 411105 China Correspondence should be addressed to Sajid Iqbal sajiduos2000@ Received 27 March 2010 Revised 2 August 2010 Accepted 27 October 2010 Academic Editor Andras Ronto 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 establish a generalization of the inequality introduced by Mitrinovic and Pecaric in 1988. We prove mean value theorems of Cauchy type for that new inequality by taking its difference. Furthermore we prove the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity. Finally we define new means of Cauchy type and prove the monotonicity of these means. 1. Introduction Let K x t be a nonnegative kernel. Consider a function u a b R where u e U v K and the representation of u is u x f K x t v t dt a for any continuous function v on a b . Throughout the paper it is assumed that all integrals under consideration exist and that they are finite. The following theorem is given in 1 see also 2 page 235 . Theorem . Let ui e U v K i 1 2 and r t 0 for all t e a b . Also let ộ R R be a function such that ộx is convex and increasing for x 0. Then cb a r x ộ u1 x u2 x b x jas x ộ 1 21 dx a 2 Journal of Inequalities and Applications where s x v2