From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Another look at a Putnam problem Date: 20 Jan 2000 23:03:15 GMT Newsgroups: sci.math Summary: [missing] During the discussion of the 1999 Putnam exam questions, many interesting responses were generated to problem B4: given a C^3 function on the real line with all derivatives positive and f'''Note that the only property of f''' we used was that f''' < f; we used >neither f''' > 0 nor f'' > 0, yet we got a constant better than 2. (He got 1.651 for the constant.) My analysis also suggested better results should be true: that perhaps { f>0, f'>0, f''' { f' <= f } At the time, I suggested a method of approach: to show that the functions under consideration admit a representation f(t) = integral( P(u) exp( u t) du, u \in [0,1] ) for some positive kernel P on the unit interval (taking a Stieltjes integral or allowing a measure with point masses or something). All these functions meet the stated conditions, and any such function satisfies the desired conclusion. This is very intriguing but cannot really be true, since it would suggest that much more is actually true: that all higher derivatives are also positive and each is less than the ones before it. Well, I never really got very far with this idea, except to generalize it a bit: For what values of (k,l,m,n) can we show that (*) { f>0, f'>0, f^(k) < f^(l) } ==> { f^(m) <= f^(n) } ? In order to construct interesting examples (and, perhaps, to prove something) I found it useful to refer to h = f'/f, a well-defined positive smooth function on the line. The ordering (and positivity) of the derivatives of f is then equivalent to the ordering (and positivity) of these functions: 1 h h' + h^2 h'' + 3 h h' + h^3 h''' + 4 h h'' + 3 h'^2 + 6 h' h^2 + h^4 ... It's easy to generate some negative examples to conjecture (*) when m > k. For example, when (k,l)=(1,0) we may take h = 1/2 + (1/pi) arctan(4 x) and discover that h'+h^2 isn't even less than 1, much less being less than h. So it seems more productive to restrict attention to (*) in cases with m <= k. (I've also looked only at k > l and m > n, in keeping with the Putnam problem.) The conjecture (*) is true in the case (k,l)=(2,1), (m,n)=(1,0), and thus when (k,l)=(2,1) and (m,n)=(2,0). I can also prove it when (k,l)=(2,0) and (m,n)=(1,0), but the other case (k,l)=(2,0) and (m,n)=(2,1) is not true; try h=1/sqrt(1+exp(-2x)). The next interesting cases are those with k=3; this includes the Putnam problem. Can anyone prove any of the various cases, or provide counterexamples? One thing is certainly true: the proposed integral respresentation _must fail_ because there are smooth functions having f''' h > h'+h^2 > h"+3hh'+h^3 but the next inequality does not hold; that is, we may construct examples with f>0, f'>0, and f''' < f'' < f' < f but f^(iv) not even less than f'', much less f''' . In particular, this f is certainly not expressible as an integral \int P(u) exp(ut) as above. I don't know what does characterize the functions representable in this way (a characterization would have to include more assumptions, e.g. that _all_ derivatives are positive). dave