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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

    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.
    Appears in Collections:[數學學系暨研究所] 學位論文

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