我们知道,当a、b、c为实数时有 (a~2+b~2)~(1/2)+(b~2+c~2)~(1/2)+(c~2+a~2)~(1/2)≥2~(1/2)|a+b+c|当且仅当a=b=c时等号成立,本文将证明这种形式的另一个更为有用的不等式,并将其加以推广。设a、b、c为实数,则下列不等式成立2(a~2+b~2+c~2)~(1/2)≤(a~2+b~2)~(1/2)+(b~2+c~2)~(1/2)++(c~2+a~2)~(1/2)≤6~(1/2)(a~2+b~2+c~2)~(1/2)。(1) 证由不等式x_1+x_2+x_3≤≤(3(X_1~2+X_2~2+X_3~2))~(1/2) We know that when a, b, c are real numbers (a~2+b~2)~(1/2)+(b~2+c~2)~(1/2)+(c~2+ a~2)~(1/2)≥2~(1/2)|a+b+c| If and only if a=b=c the equal sign holds, this article will prove that the other of this form is more Useful inequalities and promote them. Let a, b, c be real numbers, then the following inequality holds 2(a~2+b~2+c~2)~(1/2)≤(a~2+b~2)~(1/2)+ (b~2+c~2)~(1/2)++(c~2+a~2)~(1/2)≤6~(1/2)(a~2+b~2+c ~2)~(1/2). (1) The evidence from the inequality x_1+x_2+x_3≤≤(3(X_1~2+X_2~2+X_3~2))~(1/2)