| Summary: | 碩士 === 中原大學 === 應用數學研究所 === 96 === Abstract
The theory of (a,b)-convex functions was introduced by Norber Kuhn in 1987【1】.Kuhn focused mainly on the structure and the properties of (a,b)-convex functions.And we generalize a result raised by Zygfryd Kominek in 1992【2】.He would like to know on what conditions under which an (a,b)-convex function
is a constant function.
Given a function f：I→[－∞ , ∞），we define the following three sets：
(Ⅰ) K’(f)＝﹛(a,b)€(0,1)×(0,1) | f is an (a,b)-convex function﹜
(Ⅱ) K(f)＝﹛a€(0,1) | f is an a-convex function﹜
(Ⅲ) A’(f)＝﹛(a,b)€(0,1)×(0,1) | f is an (a,b)-affine function﹜
We proceed to discuss the properties of these sets K’(f)、K(f) and A’(f).Then we show that，if f：I→[－∞ , ∞）is a continuous (a,b)-convex function，f is a convex function. Finally we prove that，if (a,b)€K’(f)，a≠b and a€Q ，then f is a constant function.
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