若f:[a,b]→R為凸函數,a,b∈R,則 f(a+b/2)≤1/(b-a) ∫_a^bf(x)dx≤f(a)+f(b)/2 恆成立,這就是著名的Hermite-Hadamard雙邊不等式,要探討的是,若f在[a,b]中的凸函數,則是否存在兩實數l及L, 使得下列不等式能成立: f(a+b/2)≤l≤1/(b-a) ∫_a^bf(x)dx≤L≤f(a)+f(b)/2 本論文研究的主要目的是要對上式提供一些解。
If f:[a,b]→R is convex on [a,b] , then f(a+b/2)≤1/(b-a) ∫_a^bf(x)dx≤f(a)+f(b)/2 is known in the literature the Hermite-Hadamard inequality. There is the question that if f is a convex function on [a,b], do there exist real numbers l and L such that f(a+b/2)≤l≤1/(b-a) ∫_a^bf(x)dx≤L≤f(a)+f(b)/2 The major goal of this study is to give some answers to the question.