一些更精細的 Hermite-Hadamard 不等式

Several Refinements of Hermite-Hadamard Inequality

廖俊欣(Chun-Hsin Liao)

指導教授 ： 楊國勝

淡江大學/理學院/數學學系碩士在職專班/碩士(2018年)

https://doi.org/10.6846/TKU.2018.00369

摘要

若f: [a,b]→R為凸函數, a,b∈R ,則f((a+b)/2) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)]恆成立, 這就是著名的Hermite-Hadamard不等式, 繼續探討的是,若f為[a,b]中的凸函數, 是否能找到實數l及L使得下列不等式能成立: f((a+b)/2) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)] 本論文研究的主要目的是要對上式提供一些答案。

關鍵字

Hermite-Hadamard 不等式 ； 凸函數

並列摘要

If f: [a,b]→R is convex on [a,b], then f((a+b)/2) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)] 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) ≤ 1/(b-a)∫_a^b f(x)dx ≤ 1/(2) [f(a)+f(b)] ? The major goal of this study is to give some answers to the question.

並列關鍵字

Hermite-Hadamard inequality ； convex functions

參考文獻

[1] S. S. Dragomir and C. E. M. Pearce, Selected Topics on

Hermite-Hadamard Inequalities, (RGMIA Monographs

http: / /rgmia.vu.edu.au /monographs/ hermite_hadamardhtml),

Victoria University, 2000.

[2] A El Farissi, Simple proof and refinement of Hermite-Hadamard inequality,