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18.03.2013

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.


a1 + a2 + ... + an

n

n a1a2...an , aiR, ai ≥ 0, i = 1,2,...,n.


, ai = 0, , - . ai > 0.

. Gm = m a1a2...am. ,

am+1 = Gm+1m+1/Gmm = Gm(Gm+1/Gm)m+1 = Gm(1 + (Gm+1/Gm - 1))m+1.

Gm(1 + (Gm+1/Gm - 1))m+1Gm(1 + (m+1)(Gm+1/Gm - 1)) = Gm + (m+1)Gm+1 - (m+1)Gm = (m+1)Gm+1 - mGm.

am+1 ≥ (m+1)Gm+1 - mGm (*)

.

1. .

n = 1 a1 = a1. .

2. .

n = k ,

a1 + a2 + ... + ak

k

k a1a2...ak = Gk.

, n = k+1.

:

a1 + a2 + ... + akkGk.

(*) m = k :

a1 + a2 + ... + ak + ak+1kGk + (k+1)Gk+1 - kGk = (k+1)Gk+1.

a1 + a2 + ... + ak+1

k+1

Gk+1 = k+1 a1a2...ak+1.

, . .


, , a1 = a2 = ... = an.



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