關於Hadamard不等式一些更細緻的結果

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Title:	關於Hadamard不等式一些更細緻的結果
Other Titles:	Several refinements of Hadamard inequality
Authors:	古皓彰;Ku, Hao-Chang
Contributors:	淡江大學中等學校教師在職進修數學教學碩士學位班楊國勝;Yang, Guo-Sheng
Keywords:	Hadamard 不等式;凸函數;Hermite-Hadamard’s inequality;Convex function
Date:	2015
Issue Date:	2016-01-22 14:52:23 (UTC+8)
Abstract:	設f:Ⅰ→R是一個定義在區間Ⅰ上的凸函數，a,b∈Ⅰ a<b f(∑_(i=1)^n▒〖λ_i∙x_i 〗)≤∑_(i=1)^n▒〖λ_i∙f(x_i)〗對於所有 x_1,x_2,…x_n∈I 且 λ_1,λ_2,…λ_n∈[0,1]在 ∑_(i=1)^n▒λ_i =1則以下不等式成立 f((a+b)/2)≤1/(b-a) ∫_a^b▒〖f(x)dx≤1/2 [f(a)+f(b)] (1.1)〗 (1.1)為著名的Hadamard雙邊不等式。若f:Ⅰ→R是一個定義在區間Ⅰ上的凸函數，a,b∈Ⅰ a<b，是否存在兩個實數l,L滿足在以下的不等式 f((a+b)/2)≤l≤1/(b-a) ∫_a^b▒f(x)dx≤L≤1/2 [f(a)+f(b)] (1.2) 本論文研究的主要目的是為了提供這問題 (1.2) 更多的答案 If f : I → ℝ is convex on I, then f((a+b)/2)≤1/(b-a ) ∫_a^b▒〖f(x)dx ≤ 1/(2 ) [f(a)+f(b)] 〗 (1.1) This is the classical Hermite-Hadamard inequality If f is a convex function on I , do there exist real numbers l , L such that f((a+b)/2)≤ l ≤1/(b-a ) ∫_a^b▒〖f(x)dx ≤L ≤ 1/(2 ) [f(a)+f(b)] 〗 (1.2) The main purpose of this paper is to give more answers to the question (1.2)
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