On weighted means, geometric and arithmetic

On weighted means, geometric and arithmeti

Declaration

The integer constant $N$ satisfies

The integer dummy i is restricted to the range $0\leq i < N$

The function $p$ , with $p \in N \to \mathbb{R}$ , satisfies

(0)

For any function $g$ , with $g \in \mathbb{N} \to \mathbb{R}$ , the weighted average $w.g$ is defined by

(1) $w.q = \left\langle {\sum i : p.i * q.i}\right\rangle$

[If we so desire, we can recognize a scalar product.]

Finally, function $a$ , with $a \in N \to R$ , satisfies

$$ \langle \forall i: 0 \leq a. i \rangle $$

The well-known fact that the geometric mean is at most the arithmetic mean, i.e.,

(2) $\sqrt[N]{\langle \pi_i : a_i \rangle} \leq \langle \Sigma_i : a_i \rangle / N$

is a special case of

(3) $\langle \pi_{i} : a_{i}^{p_{i}} \rangle \leqslant \langle \Sigma_{i} : p_{i} * a_{i} \rangle$

[Take $p.i = 1 / N$ , and (3) reduces to (2).] Our target is to prove (3). Taking logarithms at

both sides, we are led to consider the relation

(4) $w_{*}(\log^{o}a)\leq \log .(w.a)$

where $0$ denotes functional composition. Because of the monotonicity of the exponential function, (4) implies (3), and our target is now to prove (4).

Consider in the plane the weights p.i placed at Q.i, with Q.i = (a.i, log.(a.i)). Their centre of gravity C is by definition and (1) (5) $C = (w.a, w.(\log a))$

We now use the fact for $p$ satisfying (0) that with weights $p.i$ placed on a convex body, their centre of gravity does not lie outside the convex body (in general it lies in-side). [Note that this is a possible definition of convexity.]

Since the second derivative of log.x is negative, the curve $y = \log x$ is convex, and hence the centre of gravity C of non-ega.

tive weights placed on the curve lies on or under the curve, more precisely on or below point B given by

$$ (6) \quad B = (w. a, \log . (w. a)) $$

Comparing their y-coordinates - see (5) and (6), we observe

$$ w. (\log^ {o} a) \leq \log . (w. a) $$

which had to be proved.

Acknowledgement Thanks to the members of the ATAC that attended yesterday's session where this problem was discussed. The above proof I designed last night. I found the theorem (3) in [0,p 102] (and prefer my proof).

[0] Aigner, Martin & Ziegler, Gunter M., "Proofs from THE BOOK", Springer-Verlag Berlin, Heidelberg, New York, 1998. [ISBN 3-540-63698-6]

Austn, 9 February 2000

prof. dr. Edsger W. Dijkstra Department of Computer Sciences The University of Texas at Austin Austin, TX 78712-1188 USA