※ 引述《tzhau (生命中無法承受之輕)》之銘言：
: 設x>=1,y>=1,z>=1,a>1 ,
: A=[(log_a x)/(1+log_a x)]+[(log_a y)/(1+log_a y)]+[(log_a z)/(1+log_a z)]
: B=(log_a xyz)/(1+log_a xyz)
: 試證A>=B ,並求等號成立的條件
: 希望有板友給點提示, 謝謝.
令f(x)=(log_a x)/(1+log_a x),x≧1
=> f(x) + f(y)
= (log_a x)/(1+log_a x) + (log_a y)/(1+log_a y)
= [(log_a x)(1+log_a y)+(log_a y)(1+log_a x)]/[(1+log_a x)(1+log_a y)]
= (log_a x+log_a y+2log_a x*log_a y)/(1+log_a x+log_a y+log_a x*log_a y)
= (log_a xy+2log_a x*log_a y)/(1+log_a xy+log_a x*log_a y) ...(1)
≧ (log_a xy)/(1+log_a xy) ...(2)
= f(xy)
=> f(x) + f(y) + f(z) ≧ f(xy) + f(z) ≧ f(xyz)
"="成立 <=> x = y = 1 & xy = z = 1 <=> x = y = z = 1
註(1)
因為x,y≧1,a>0 => log_a x,log_a y≧0
(log_a xy+2log_a x*log_a y)(1+log_a xy)-(1+log_a xy+log_a x*log_a y)(log_a xy)
= 2log_a x*log_a y + log_a x*log_a y*log_a xy ≧ 0
註(2)
"="成立 <=> log_a x*log_a y = 0 <=> log_a x = log_a y = 0 <=> x = y = 1
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◆ From: 111.252.217.131
※ 編輯: mack 來自: 111.252.217.131 (02/09 20:30)