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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 On Logarithmic Convexity for Power Sums and Related Results | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 389410 9 pages doi 10.n55 2008 389410 Research Article On Logarithmic Convexity for Power Sums and Related Results J. PeCariC1 2 and Atiq Ur Rehman2 1 Faculty of Textile Technology University of Zagreb 10000 Zagreb Croatia 2Abdus Salam School of Mathematical Sciences GC University Lahore 54660 Pakistan Correspondence should be addressed to Atiq Ur Rehman mathcity@gmail.com Received 28 March 2008 Revised 23 May 2008 Accepted 29 June 2008 Recommended by Martin j. Bohner We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results. Copyright 2008 J. Pecaric and A. U. Rehman. 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 preliminaries Let x x1 . xn p P1 . pn denote two sequences of positive real numbers with S 1 Pi 1. The well-known Jensen Inequality 1 page 43 gives the following for t 0 or t 1 n ìrì -i 1 n t TjPixi i 1 1.1 and vice versa for 0 t 1. Simic 2 has considered the difference of both sides of 1.1 . He considers the function defined as i sn x - n 1piXi t t t - 1 log Ệ Pixi tpx log xi i 1 t f 0 1 n Pi log xi i 1 n n yPixA log i ypixA t 1 t 0 1.2 and has proved the following theorem. 2 Journal of Inequalities and Applications Theorem 1.1. For -TO r s t TO then X r Xr y-s Xty-r. 1-3 Anwar and PeCariC 3 have considerd further generalization of Theorem 1.1. Namely they introduced new means of Cauchy type in 4 and further proved comparison theorem for these means. In this paper we will give some results in the case where instead of means we have power sums. Let x be positive n-tuples. The well-known inequality for power sums of order s and r for
