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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 Stolarsky Type Means | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 720615 15 pages doi 10.1155 2010 720615 Research Article Generalization of Stolarsky Type Means J. Pecaric1 2 and G. Roqia2 1 Faculty of Textile Technology University of Zagreb Pierottijeva 6 10000 Zagreb Croatia 2 Abdus Salam School of Mathematical Sciences GC University Lahore 54000 Pakistan Correspondence should be addressed to G. Roqia rukiyya@gmail.com Received 27 April 2010 Revised 10 August 2010 Accepted 15 October 2010 Academic Editor Paolo E. Ricci Copyright 2010 J. Pecaric and G. Roqia. 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 generalize means of Stolarsky type and show the monotonicity of these generalized means. 1. Introduction and Preliminaries The following double inequality is well known in the literature as the Hermite-Hadamard H.H integral inequality f a b 1 b C v rtvsf A f b 2 - b-a. J. f Mdx - 2 1.1 provided that f a b R is a convex function 1 page 137 2 page 1 . This result for convex functions plays an important role in nonlinear analysis. These classical inequalities have been improved and generalized in a number of ways and applied for special means including Stolarsky type logarithmic and p-logarithmic means. A generalization of H.H inequalities was obtained in 3-5 2 page 5 and 1 page 143 . Theorem 1.1. Let p q be positive real numbers and a1 a b bi be real numbers such that a1 - a b - b1. Then the inequalities fCf-fib - pfMdx -pfwfqf b p q 2y A-y p q 1.2 2 Journal of Inequalities and Applications hold for A pa qb p qf y 0 and all continuous convex functions f a1 b1 R ifandonlyif y b - a p q min p q . 1.3 Remark 1.2. The inequalities given by 1.2 are strict if f is a continuous strictly convex on a1 b1 . If we keep the assumptions as stated in Theorem 1.1 we also have 1 page 146
