臺灣博碩士論文加值系統

設f：[a,b]→R是一個定義在[a,b]的凸函數，且a,b∈R，則
f((a+b)/2)≤1/(b-a) ∫_a^bf(x)dx≤1/2[f(a)+f(b)]恆成立-(1.1)
上式(1.1)為著名的阿達瑪雙邊不等式。
要探討若f是在[a,b]中的凸函數，則是否存在兩實數l,L使得
f((a+b)/2)≤l≤1/(b-a) ∫_a^bf(x)dx≤L≤1/2[f(a)+f(b)] -(1.2)本論文研究主要目的是為了能在上式(1.2)中提出一些解法。

If f：[a,b]→R is convex on [a,b],then
f((a+b)/2)≤1/(b-a) ∫_a^bf(x)dx≤1/2[f(a)+f(b)](1.1)
This is the classic Hermite-Hadamard inequality
The question is that If f is convex function on [a,b],do there exist real numbers l,L,such that
f((a+b)/2)≤l≤1/(b-a) ∫_a^bf(x)dx≤L≤1/2[f(a)+f(b)](1.2)
The main purpose of this paper is to give some answers to the question.(1.2)

1.Introduction………………………………………………1
2.Main Results………………………………………………4
Reference……………………………………………………17

[1] M.BESSENYEI AND Zs. PÁLES,Higer-order generalizations of Hadamard’sinequality,Publ.Math.Debrecen,61,3-4 (2002),623-643.
[2] M.BESSENYEI AND Zs. PÁLES,Hadamard-type inequality for generalized convex functions,Math.Inequal. Appl.,6,3 (2003),379-392.
[3] S. S. DRAGOMIR AND C. E. M. PEARCE,Selected Topics on Hermite-Hadamard Inequalities,
(RGMIA Monographs http://rgmia.vu.edu.au/monographs/hermite_hadamard.html),
VictoriaUniversity, 2000.
[4] A.EL FARISSI,Z LATREUCH, B BELAIDI, Hadamard-Type Inequalities for Twice DifferentibleFunctions,RGMIA Research Report collection, 12, 1 (2009),Art. 6.
[5] A.EL FARISSI,Simple proof and refinement of Hermite-Hadamardinequality,J.MathInequal Vol.4 No.3 (2010),365,369.
[6] A. M. FINK,AbestpossibleHadamard’sinequality,Math. Inequal. Appl., 1, 2 (1998), 223-230.
[7] J. HADAMARD,Étude sur les propriétés des fonctionsentières et enparticulierd’unefonctionconsidérée par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.
[8] D. S. MITRINOVIĆ AND I. B. LACKOVIĆ,Hermite and convexity, Aequationes Math., 28 (1985), 229-232.
[9]C. NICULESCU AND L.-E.PERSSON,Old and new on the Hermite-Hadamard inequality, Real Analysis Exchange, 2004.
[10] Carlem Wang(王清正),Refinements of Hermite-Hadamard inequality (2016)
[11]Chi-Heng Chien(簡杞恆),More advancement of Hermite-Hadamard Inequality(2010)