Given the assumptions and notation of Problem 1 above, show that 4∆7∆9 ≤ ∆2 8 [15]. This inequality puts an additional constraint on

∆7, ∆8, and ∆9 besides the obvious nonnegativity requirements and the sum requirement ∆7 + ∆8 + ∆9 = 1.

(Hints: Note first that

Φij = 1 2

∆7 +

1 4

∆8

= 1 4

Φkm +

1 4

Φkn +

1 4

Φlm +

1 4

Φln.

Next apply the inequality (a + b)2 ≥ 4ab to prove 4∆7 ≤ (4Φij )2;
finally, rearrange.)