Refinements of Inequalities about Power Means

碩士論文[[abstract]]令n為正整數，a(i)>0 (i=1,…,n)且 a(1)+…+a(n)=1。令 x(i)>0 (i=1,…,n)記 M(r), m(r)分別為x(1),…,x(n)的加權及未加權次方平均數。最近幾年來許多有關此二平均數之不等式被發表[1-5,7-8]。下列定理是有關次方平均數的古典不等式。 定理. 令r, s皆為實數且r M(r)< or = M(s) (1.1) 此不等式等號成立若且唯若x(1)=…=x(n)。 本論文我們將找到一些連續、嚴格r-凸函數和連續嚴格r-凹函數分別細分不等式(1.1)且推廣Chong[4]和Yang-Wang[8]不等式。[[abstract]]Let n be a positive integer and a(i)>0 (i=1,…,n) with a(1)+…+a(n)=1. Let x(i)>0 (i=1,…,n) and let M(r) and m(r) be the weighted and unweighted power means on x(1),…,x(n), respectively. In recent years many interesting inequalities involving these means haves been published see, for examples, [1-5] and [7-8]. A classical inequality for the power means is the following theorem. Theorem