欧拉不等式的广义积分插入(英文)

从所周知,欧拉不等式2r≤R2(3)~(1/3)r≤3~(1/3)R。(1765)我们可加细到2(3)~(1/3)r≤(abc)1/3≤1/3(a b c)≤3~(1/3)R;(1)2(3)~(1/3)r≤(abc)~(1/3)≤{P integral from n=1 to ∞( 8)(a x)(b x)(c x)]~-(P 1)3dx}-1/P≤1/3(a b c)≤3~(1/3)R;(2)2(3)~(1/3)≤(abc)~(1/3){P integral from n=1 to ∞( 8)(a x)(b x)(c x)]~-(P 1)/3dx}~-(1/P)≤{Pintegral from n=1 to ∞( 8)λ~(-1)(ι λ)(a x))~(1/3)(ι λ(b x))~(1/3)(ι λ(c x))~(1/3)-ι]~(-P-1)dx}~(-1/P)≤1/3(a b c)≤3~(1/3)R。(3)

Euler's Inequality of Insertion Improper Integral