已知a、b、c为正数,n是正整数,且f(n)=lgan+bn+cn3,求证:2f(n)≤f(2n).

问题描述:

已知a、b、c为正数,n是正整数,且f(n)=lg

an+bn+cn
3
,求证:2f(n)≤f(2n).

证明:∵a2+b2≥2ab∴(an+bn+cn)2=a2n+b2n+c2n+2an•bn+2an•cn+2bn•cn≤3(a2n+b2n+c2n)∴lg(an+bn+cn)2≤lg[3(a2n+b2n+c2n)]∴lg(an+bn+cn)2≤lg(a2n+b2n+c2n)+lg3∴2lg(an+bn+cn)≤lg(a2n+b2n+c2n...