Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where every derivative of $g$ is strictly monotonic? If not, what condition should we use so that the continuation is unique but other than requiring $f$ to be analytic?
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$\begingroup$ At first, one may have a knee-jerk reflex about smooth non-zero functions which have all derivative $0$ at $0$, and the related functions. It's not quite that simple thought. $\endgroup$– Włodzimierz HolsztyńskiCommented Jan 18, 2017 at 17:39
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1$\begingroup$ Why the votes to close? This seems a very reasonable question to me. (After the edits anyway.) $\endgroup$– Todd TrimbleCommented Jan 19, 2017 at 11:50
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$\begingroup$ What about $f(x)=1/x$ (and $a=-1$)? $\endgroup$– ACLCommented Jan 20, 2017 at 23:10
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$\begingroup$ @ACL: OP does not ask about existence but about uniqueness. $\endgroup$– András BátkaiCommented Jan 21, 2017 at 7:30
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EDITED. The following theorem of Bernstein answers the question:
If $f$ is infinitely differentiable on an interval and no derivative changes sign, then $f$ is analytic.
Your condition that all derivatives are monotone of course implies that none of them changes sign. Therefore, if such a function is extended on a larger interval with preservation of the property that no derivative changes sign, then such an extension is unique.
S. Bernstein, Sur la définition et les propriétés des fonctions analytiques d'une variable réelle, Math. Ann. vol. 75 (1914) pp. 449-468.
Here is a link: http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00226580X The theorem is stated in section 5. Actually it is much stronger than I stated.
A survey of the later results on the topic is
Polya, G. On the zeros of the derivatives of a function and its analytic character. Bull. Amer. Math. Soc. 49, (1943). 178–191.
answered Jan 19, 2017 at 7:51
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$\begingroup$ Your comment is interesting, but I require all derivatives strictly monotone - it does not mean all derivatives non- negative. (e.g. ln(x) ). $\endgroup$ Commented Jan 20, 2017 at 10:35
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$\begingroup$ Also, there is a question whether a smooth continuation is unique if we require all derivatives to be strictly monotonic. $\endgroup$ Commented Jan 20, 2017 at 10:49
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$\begingroup$ exp(-x) is a better example of a smooth function whose all derivatives are strictly monotonic but some are negative, some positive. $\endgroup$ Commented Jan 20, 2017 at 14:38
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$\begingroup$ @Tomasz Grzybowski: I edited my answer. $\endgroup$ Commented Jan 20, 2017 at 19:23
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1$\begingroup$ There is also a Russian translation in Bernstein's Collected papers (also available on I-net) for those who do not read French. $\endgroup$ Commented Jan 20, 2017 at 23:01