Problem. Let k be a positive integer. f:[0,∞)→[0,∞) is a continuous function such that f(f(x))=xk, ∀x∈[0,∞). Show that ∫10f(x)2dx≥2k−1k2+6k−3.This statement is provable by Young's inequality. By assuming f be continuously differentiable (or any kind of continuous such that the integration by parts on [0,1] work, say f is absolutely continuous), then we have a stronger bound: ∫10f(x)2dx≥2k−1k2+4k−2. Which becomes the previous one by adding 2k−1 to the denominator.
Solution. There is a few observations. First, f is injective. Second, f(f(x))=xk implies f(f(f(x)))=f(x)k, therefore f(xk)=f(x)k. In particular, if we put x=0 and x=1, then we have f(0)=f(0)k and f(1)=f(1)k. Therefore both f(0),f(1)∈{0,1}. By injectivity and the fact f≥0, we have f(0)=0 and f(1)=1.
By Cauchy-Schwarz inequality we have I:=∫10f2dx≥∫10fdx. We substitute x=f(y) to get I=∫10f(f(y))(f(y))′dy=∫10yk(f(y))′dy=1−k∫10f(y)yk−1. The last equality follows from integration by parts. Now we shrink RHS by Cauchy-Schwarz inequality to get I≥1−k√2k−1√I, which we rearrange to get (√I)2+k√2k−1√I−1≥0. By completing square we find that this implies √I≥−k√2k−1+√k2+8k−42k−12=2√2k−1√k2+8k−4+k. Now by Cauchy-Schwarz inequality we also have √k2+8k−4+k≤√(k2+8k−4+k2)(1+1)=2√k2+4k−2, hence we have I≥2k−1k2+4k−2.◼
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