对数平均的推广(英文)

GENERLIZATIONS OF THE LOGARITHMIC MEAN

记J_t(x,y)=[t(x^(t+1)-y^(t+1))]/[(t+1)(x^t-y^t)]。它有性质:J_(-1/2)2(x,y)=G(x,y),J_(1/2)(x,y)=He(x,y),J_1(x,y)=A(x,y)。我们证明了J_1(x,y)关于t单调增加。同时有(?)J_t(x,y)=L(x,y)。那么我们有不等式G(x,y)≤L(x,y)≤He(x,y)≤A(x,y)。

Write J_t(x,y)=[t(x^t+1-y^t+1)]/[(t+1)(x^t-y^t)].This has property J_-1/2(x,y) =G(x,y),J_1/2(x,y)=He(x,y),J_1(x,y)=A(x,y).We prove that it increases with t.Also Lir J_1(x,y)=L(x,y).So we have the inequalities G(x,y)≤L(x,y) t→o ≤He(x,Y)≤A(x,y).