| Summary: | 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.
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