Weighted Means

Let a₁, a₂, ……, a_n be n positive real numbers and m₁, m₂, …, m_n 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+...+m_na_n/m₁+m₂+...m_n, G* = (a₁^m₁.a₂^m₂ ...... a_n^m_n)1/(m₁+m₂+...+m_n) and H* = m₁+m₂+...m_n/(m₁/a₁+m₂/a₂+...+m_n/a_n).

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

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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