=+5.34. Prove Carlsons inequality: If f 0 on [0, ), then 0 f(x) dx
Question:
=+5.34. Prove Carlson’s inequality: If f ≥ 0 on [0, ∞), then
∞
0 f(x) dx ≤ √π
∞
0 f 2(x) dx 1/4 ∞
0 x2f 2(x) dx 1/4
.
[Hint: For any
a, b > 0, write
∞
0 f(x) dx =
∞
0 1
√a + bx2 a + bx2f(x) dx.
5.7 Exercises 169 Also, note that ∞
0 (a + bx2)−1 dx = π/2
√
ab. ]
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