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Weighted Means

Let a1, a2, ……, an be n positive real numbers and m1, m2, …, mn be n positive rational numbers. Then we define weighted Arithmetic Mean (A*), weighted Geometric Mean (G*) and weighted Harmonic Mean (H*) as

A* = m1a1+m2a2+...+mnan/m1+m2+...mn, G* = (a1m1.a2m2 ...... anmn)1/(m1+m2+...+mn) and H* = m1+m2+...mn/(m1/a1+m2/a2+...+mn/an).

It can be shown that A* > G* > H*. Moreover equality holds at either place if and only if a1 = a2 = … = an.

Illustration:

If a, b, c are positive real numbers such that a + b + c = 18, find the maximum value of a2b3c4.

Solution:

        a + b + + c = 18  2.a/2 + 3.b/3 + 4.c/4 = 18

        ⇒ ((a/2)2.(b/3)3.(c/4)4)1/9 < 1/9 (2.a/2 + 3.b/3 + 4.c/4)

         a2b2c4 < 29.22.33.44.

        Thus the maximum value of a2 b3 c4 is 42 . 63 . 84 = 219.33.

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