cho a,b,c >0. CMR $\frac{a^{n}+b^{n}+c^{n}}{3}\geq \left ( \frac{a+b+c}{3} \right )^{n}$
$\frac{a^{n}+b^{n}+c^{n}}{3}\geq \left ( \frac{a+b+c}{3} \right )^{n}$
Started By thangnhoc9x, 02-03-2013 - 19:34
#1
Posted 02-03-2013 - 19:34
- nguyen tien dung 98 likes this
#2
Posted 02-03-2013 - 19:39
cho a,b,c >0. CMR $\frac{a^{n}+b^{n}+c^{n}}{3}\geq \left ( \frac{a+b+c}{3} \right )^{n}$
Ta cần chứng minh
$3^{n-1}(\sum a^{n})\geq (\sum a)^{n}$
Thật vậy, theo bất đẳng thức $Holder$ ta có
$3^{n-1}\sum a^{n}=(1+1+1)(1+1+1)...(1+1+1)(a^{n}+b^{n}+c^{n})\geq (a+b+c)^{n}$
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#3
Posted 02-03-2013 - 19:45
ko làm holder được không?
#4
Posted 02-03-2013 - 20:01
Chứng minh bằng quy nạp. ( Mình tắt bước đầu nhé).ko làm holder được không?
Ta cần chứng minh: $\frac{a^{k+1}+b^{k+1}+c^{k+1}}{3}\geq (\frac{a+b+c}{3})^{k+1}$
Hay tương đương với chứng minh: $\frac{a^{k+1}+b^{k+1}+c^{k+1}}{3}\geq (\frac{a+b+c}{3})(\frac{a^k+b^k+c^k}{3})\Leftrightarrow 3(a^{k+1}+b^{k+1}+c^{k+1})\geq (a+b+c)(a^k+b^k+c^k)\Leftrightarrow 2(a^{k+1}+b^{k+1}+c^{k+1})-a(b^k+c^k)-b(c^k+a^k)-c(a^k+b^k)\geq 0\Leftrightarrow (a-c)(a^k-c^k)+(b-a)(b^k-a^k)+(b-c)(b^k-c^k)\geq 0$ (Xong)
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