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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 General Convexity of Some Functionals in Seminormed Spaces and Seminormed Algebras | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 643768 6 pages doi 10.1155 2010 643768 Research Article General Convexity of Some Functionals in Seminormed Spaces and Seminormed Algebras Todor Stoyanov Department of Mathematics Varna University of Economics Boulevard Knyaz Boris 177 Varna 9002 Bulgaria Correspondence should be addressed to Todor Stoyanov todstoyanov@yahoo.com Received 30 July 2010 Accepted 27 October 2010 Academic Editor S. S. Dragomir Copyright 2010 Todor Stoyanov. 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 prove some results for convex combination of nonnegative functionals and some corollaries are established. 1. Introduction Inequalities have been used in almost all the branches of mathematics. It is an important tool in the study of convex functions in seminormed space and seminormed algebras. Recently some works have been done by Altin et al. 1 2 Tripathy et al. 1-6 Tripathy and Sarma 3 4 Chandra and Tripathy 5 Tripathy and Mahanta 6 and many others involving inequalities in seminormed spaces and convex functions like the Orlicz function. In this paper inequalities for convex combinations of functionals satisfying conditions a and b are formulated in the theorems and some corollaries are proved using the theorems. Condition a relates to nonnegative functionals over which the inequalities in Theorems 1.1 and 1.4 on seminorm are proved. In Theorem 1.1 we consider seminormed spaces and in Theorem 1.4 seminormed algebras. Condition b relates generally to the representations between seminormed spaces and seminormed algebras. The inequalities formulated in this way are proved in Corollaries 1.2 and 1.5. In this paper we consider the following generalization of the convexity in seminormed algebras. A Yf 2m1 PiXi Zm Wpi f Xi where m1 .
