Loading [MathJax]/jax/output/HTML-CSS/jax.js

Wednesday, December 11, 2013

Integral Inequality

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)2dx2k1k2+6k3.
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)2dx2k1k2+4k2. Which becomes the previous one by adding 2k1 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 f0, we have f(0)=0 and f(1)=1.

By Cauchy-Schwarz inequality we have I:=10f2dx10fdx. We substitute x=f(y) to get I=10f(f(y))(f(y))dy=10yk(f(y))dy=1k10f(y)yk1. The last equality follows from integration by parts. Now we shrink RHS by Cauchy-Schwarz inequality to get I1k2k1I, which we rearrange to get (I)2+k2k1I10. By completing square we find that this implies Ik2k1+k2+8k42k12=22k1k2+8k4+k. Now by Cauchy-Schwarz inequality  we also have k2+8k4+k(k2+8k4+k2)(1+1)=2k2+4k2, hence we have I2k1k2+4k2.

No comments:

Post a Comment