Unknown

7.3. Inequalities 221

2. (a) Verify the identity

x6 + y6+z6+u6+~6+w6-6xyzuvw

= ~(x2+y2+,a)[(y2-z2)2+(%2-x2)2+(x2-y2)2]

+ i(u2 + v2 + w2)[(v2 - w2)2 + (w2 - u2)2 + (u2 - v2)2]

+ 3(xyt - uvw)2.

(b) Establish the AGM inequality for n = 6.

3. Suppose it is known that, for a certain positive integer k,

x:+x;+x;+--+x; 2 kxlxz.*.xk

for any nonnegative real xi. Prove that

YT + Y2” + Y3” +.. - + YE 2 vm... yfl

for any nonnegative real yi, where n = 2k or n = 3k.

4. Deduce from Exercise 3 that the AGM inequality holds for n = 2’3’
   for T, s nonnegative integers.


5. We wish to demonstrate for any positive integer n and real variables
   xi that
   xy + x; +.. * +xi -nx122...x, 2 0.
   One can follow the procedure suggested by Exercise 4 in which the
   problem can be reduced to the case in which n is prime. However, for
   larger values of n, it is far from clear whether one can conveniently
   manipulate the left side into a form which is clearly nonnegative (Ex-
   ploration E.61 asks you to investigate the case of n = 5). As often
   happens in mathematics, a more convenient argument can be found
   if we are prepared to prove a more general result. Accordingly, we in-
   troduce weights wi. These are positive real numbers wi which satisfy

The generalized arithmetic mean with weight w is defined by the

expression

w1a1+ W2Q2 + ---+ wnun

and the generalized geometric mean with weight w by

uyJlay~.. .a;“.

(a) What values of wi will yield the standard means?