r-凸函數的平均值

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Title:	r-凸函數的平均值
Other Titles:	Mean values of r-convex functions
Authors:	蕭博文;Hsiao, Po-Wen
Contributors:	淡江大學數學學系碩士班陳功宇
Keywords:	r-凸函數;阿達瑪不等式;r-convex function;Hadamard's inequlity
Date:	2013
Issue Date:	2013-04-13 11:07:39 (UTC+8)
Abstract:	若f為區間I上的連續正函數，且a,b∈I ，本文研究兩個函數 H(a,b;t)=frac{1}{b-a}int_{a}^{b}f(tx+(1-t)frac{a+b}{2})dx 與 F(a,b;t)=frac{1}{(b-a)^2}int_{a}^{b}int_{a}^{b}f(tx+(1-t)y)dxdy 我們的結果為 (1)若r≦1且f為r-凸函數，則對於所有a,b∈I，H(a,b;t)為t的 r-凸函數。 (2)若r≦1且f為r-凸函數，則對於所有a,b∈I，F(a,b;t) 為t的 r-凸函數。 (3)若對於所有a,b∈I，H(a,b;t)在[0,1]上為t的r-凸函數，則f在 I為凸函數。 (4)若對於所有a,b∈I，F(a,b;t)在[0,1]上為t的r-凸函數，則f在 I為凸函數。 For a continuous positive function f on interval I and a,b∈I, we consider two functions H(a,b;t)=frac{1}{b-a}int_{a}^{b}f(tx+(1-t)frac{a+b}{2})dx and F(a,b;t)=frac{1}{(b-a)^2}int_{a}^{b}int_{a}^{b}f(tx+(1-t)y)dxdy The followings are our results (1)If r≦1 and f is r-convex function then H(a,b;t) is r-convex function in t for all a,b in I. (2)If r≦1 and f is r-convex function then F(a,b;t) is r-convex function in t for all a,b in I. (3)If H(a,b;t) is r-convex function in t on [0,1] for all a,b in I, then f is r-convex function on I. (4)If F(a,b;t) is r-convex function in t on [0,1] for all a,b in I, then f is r-convex function on I.
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