a, b, c, A, B, C are reals with a, A non-zero such that |ax2 + bx + c| ≤ |Ax2 + Bx + C| for all real x. Show that |b2 - 4ac| ≤ |B2 - 4AC|.
Solution
Put f(x) = ax2 + bx + c, F(x) = Ax2 + Bx + C, D = B2 - 4AC, d = b2 - 4ac. wlog a, A > 0. Considering large x, it is clear that a ≤ A.
We consider separately the three cases D > 0, D = 0 and D < 0.
Case 1. D > 0. Then F has two distinct roots α, β, so F(x) = A(x-α)(x-β). But then α and β must also be roots of f, so f(x) = a(x-α)(x-β). Hence 0 < d = a2(α-β)2 ≤ A2(α-β)2 = D. So |d| ≤ |D|.
Case 2. D = 0. So F(α) = F'(α) = 0 for some α. Hence also f(α) = 0. But F(α+ε) = O(&epsilon2), so f(α+ε) = O(&epsilon2) also. Hence f '(α) = 0 also. So f also has a repeated root, so d = 0.
Case 3. D < 0. Then F(x) = A(x + B/2A)2 + C-B2/4A > 0 for all x. So we have (A-a)x2 + (B-b)x + (C-c) ≥ 0 for all x. If A = a, then it follows that B = b also. But now |d| = 4A|f(-B/2A)| ≤ 4A|F(-B/2A)| = |D|. So assume A > a. Similarly, we also have (A+a)x2 + (B+b)x + (C+c) ≥ 0 for all x. Since these two quadratics are never negative we have (B-b)2 ≤ 4(A-a)(C-c) and (B+b)2 ≤ 4(A+a)(C+c). Adding, B2 + b2 ≤ 4AC + 4ac. Hence d ≤ -D = |D|. As before, taking x = -B/2A we get -d ≤ 4af(-B/2A) ≤ 4Af(-B/2A) ≤ 4A|F(-B/2A)| = |D|.
© John Scholes
jscholes@kalva.demon.co.uk
8 Dec 2003
Last corrected/updated 8 Dec 03