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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 New Means of Cauchy’s Type | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 163202 10 pages doi 10.1155 2008 163202 Research Article New Means of Cauchy s Type Matloob Anwar1 and J. PeCariC1 2 1Abdus Salam School of Mathematical Sciences GC University Lahore Gulberg 54660 Pakistan 2 Faculty of Textile Technology University of Zagreb 10000 Zagreb Croatia Correspondence should be addressed to Matloob Anwar matlooKt@yahoo.com Received 30 December 2007 Accepted 7 April 2008 Recommended by Wing-Sum Cheung We will introduce new means of Cauchy s type Msrl f y defined for example as Msrl f y fl l - s r r - sf Mrr f y - Mr f y Ml f y - Mls f y 1 r-l in the case when l fr fs l r 0. We will show that this new Cauchy s mean is monotonic that is the following result. Theorem. Let t r u v E R such that t v r u. Then for Mf f y one has MSr MV u. We will also give some related comparison results. Copyright 2008 M. Anwar and J. Pecaric. 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 Let Q be a convex set equipped with a probability measure y. Then for a strictly monotonic continuous function f the integral power mean of order r E R is defined as follows f fừ Ỵ dy ùẦ r 0 Q Mr ự yì exp log f fuf dyfù r 0. Q 1.1 Throughout our present investigation we tacitly assume without further comment that all the integrals involved in our results exist. The following inequality for differences of power means was obtained see 1 Remark 8 r r - s l l - s m Mf - Mrs f y Ml f y - Ms f y r r - s l l - s M 1.2 2 Journal of Inequalities and Applications where r l s R l r s r l 0 and where m and M are respectively the minimum and the maximum values of the function xr - on the image of f u u Q . Let us note that 1.2 was obtained as consequence of the following result see e.g. 1 Corollary 1 . Theorem 1.1. Let
