Mean values of r-convex functions

• Main Authors: Po-Wen Hsiao, 蕭博文
• Other Authors: 陳功宇
• Format: Others
• Language: zh-TW
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/20380956530063860765

碩士 === 淡江大學 === 數學學系碩士班 === 101 === For a continuous positive function f on interval I and a,b∈I, we consider two functions H(a,b;t)=frac{1}{b-a}int_{a}^{b}f(tx+(1-t)frac{a+b}{2})dx and F(a,b;t)=frac{1}{(b-a)^2}int_{a}^{b}int_{a}^{b}f(tx+(1-t)y)dxdy The followings are our results (1)If r≦1 and f is r-convex function then H(a,b;t) is r-convex function in t for all a,b in I. (2)If r≦1 and f is r-convex function then F(a,b;t) is r-convex function in t for all a,b in I. (3)If H(a,b;t) is r-convex function in t on [0,1] for all a,b in I, then f is r-convex function on I. (4)If F(a,b;t) is r-convex function in t on [0,1] for all a,b in I, then f is r-convex function on I.