May 1st, 2007, 01:16 AM  #1 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  Bounded derivatives
Here's a cute problem. Let f: R>R be a mapping which is N+1 times differentiable, and such that both f and f^(N+1) (the N+1th derivative of f) are bounded on R. Show that all derivatives f^(k) of f are bounded on R. R designates the field of all real numbers. 
October 30th, 2007, 08:57 AM  #2 
Newbie Joined: Oct 2007 From: Brazil Posts: 8 Thanks: 0 
You're assuming f^k exists on R for every k, right? This is not part of the conclusion, right? If, for example, n =1, then the fact that f is twice differentiable and f and f'' are bounded on R does not impliy f has derivatives of all orders on R. I'm confused here. Artur 
October 30th, 2007, 09:47 AM  #3 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7 
No, what I meant is to prove that given the hypotheses, the result holds for every k<=N+1. Of course, f does not end up being infinitely differentiable.

October 30th, 2007, 10:48 AM  #4 
Senior Member Joined: Oct 2007 From: France Posts: 121 Thanks: 1 
Can we suppose that n=N?

October 30th, 2007, 12:47 PM  #5 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7 
Sorry, I edited the original message. Actually there's only N.

November 1st, 2007, 08:35 AM  #6 
Newbie Joined: Oct 2007 Posts: 1 Thanks: 0 
Well it is true for N=1... thats all i can show for now, dunno how to expand it to N=n. First if f is N+1 times differentialable on R f^k is continous everywhere. so if any derivate is unbounded its at x>>1 okay... if f is bounded then f(x)<M for all x in R thus given any 2 elements in R: a and b, where a<b i) 2M>f(a)f(b)=f'(c)(ba) for some c contained in (a,b) {meanvaluetheorem continous functions} if f' is unbounded, then given any £ > 0 there exists a real number d in R such that f'(d) > 1/£^2 Now consider the interval (d£,d+£) by i) there exists a q contained in the interval where f'(q)<2M*2£ Meanvalue theorem applied again yields f'(d)f(q)=f''(k)*(qd) for some k contained in (q,d) or (d,q) who have length < £ so f''(k)=(f'(d)f(q))/(qd)> (1/£^2  2M*2£) / £ which gets arbitrary large ie we have shown the existance of a such k for every chosen £, thus f''(k) must be unbounded (contradiction, so f' is unbounded is false) 

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