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.

第一章 前言...............................................  1
第二章 r-凸函數平均值的映射...............................  5
第三章 兩個關於導數為r-凸函數的不等式..................... 12
第四章 討論............................................... 17
參考文獻.................................................. 19

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