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Báo cáo hóa học: " Research Article On the Monotonicity and Log-Convexity of a Four-Parameter Homogeneous Mean"

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 the Monotonicity and Log-Convexity of a Four-Parameter Homogeneous Mean | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 149286 12 pages doi 10.1155 2008 149286 Research Article On the Monotonicity and Log-Convexity of a Four-Parameter Homogeneous Mean Zhen-Hang Yang Electric Grid Planning and Research Center Zhejiang Electric Power Test and Research Institute Hangzhou 310014 China Correspondence should be addressed to Zhen-Hang Yang yzhkm@163.com Received 13 April 2008 Accepted 29 July 2008 Recommended by Sever Dragomir A four-parameter homogeneous mean ĩ p q r s a b is defined by another approach. The criterion of its monotonicity and logarithmically convexity is presented and three refined chains of inequalities for two-parameter mean values are deduced which contain many new and classical inequalities for means. Copyright 2008 Zhen-Hang Yang. 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 The so-called two-parameter mean or extended mean between two unequal positive numbers x and y was defined first by Stolarsky 1 as E r s x y s xr - y 1 r-s yr xs - ys y xr - yr r ln x - ln y 1 r xs - ys s ln x - ln y 1 s r s rs Ỷ 0 r 0 s 0 r 0 s 0 1.1 xr ln x - yr ln y 1 exn xr - yr-r r s 0 r s 0. 2 Journal of Inequalities and Applications It contains many mean values for instance E 1 0 x y L x y - lnx - lny x fiy 1.2 E 1 1 x y I x y - x x y e- x 1 x- . x y y X x y 1.3 E 2 1 x y A x y x 2 y 1.4 3 1 x Axy y E 2 2 x h x y 3 . 1.5 The monotonicity of E r s x y has been researched by Stolarsky 1 Leach and Sholander 2 and others also in 3-5 using different ideas and simpler methods. Qi studied the log-convexity of the extended mean with respect to parameters in 6 and pointed out that the two-parameter mean is a log-concave function with respect to either parameter r or s on interval 0 TO and is a log-convex function on interval .