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222 7. Approximations and Inequalities

(b) Let u = w,-r/(w,-1 + wn), v = w,/(w,-1 - w,) and a =

uu,,-1 + vu,. Verify that

wlul + 20202 +. f. + w,u, = wlul + w2a2 +. f. + w,-2an-2

+ (w-1 + w,)u.

(c) Prove, by induction on n, that for all nonnegative real ai and

positive weights wi,

a1 wl a2 WC2... axm^5 wla2 +... + wnun.

6. (Knowledge of calculus required.) With the help of some simple prop-
   erties of the graph of the logarithm function, a short and general proof
   of the AGM inequality is possible. We use the notation of Exercise 5.
   For x > 0, let log x denote the logarithm of x to base e. Observe that
   D(logx) = l/x > 0 and 02(logz) = -l/z2 < 0,so that the graph of
   the equation y = logx is increasing and concave.

(a) Sketch the graph of the equation y = logt. Verify that the

tangent to the graph through the point (1,0) has the equation

y = z - 1 and that. the graph lies below its tangent through the

whole domain of x. Deduce that logz 5 x - 1 for x > 0.

(b) Verify that

log(a~la;a. *. a;* )=wlloga~+*~~+w,loga,.

(c) Let m = wiai + wzaz +. .. + ~,,a,. Show that

log&-lOgm<(Ui/?Yh)-1 (i=1,2,...,n).

(d) By multiplying the ith equation in (c) by wi and adding, show

that

(CWi log Ui) - log ?73 5 0.

Deduce from this the generalized AGM inequality.

7. Prove the following

(a) x+x-l>2 forx>O

(b) 4x(1 - Z) 5 1 for + E R.

8. Let al,... , a, be positive reals. Use the CSB inequality to establish
   that
   (a~+u2+~~~+u,)(u~1+u~1+.--+u~‘)>n2
   with equality iff all the oi are equal.