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?

  • $\begingroup$ 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 Holsztyński Jan 18 '17 at 17:39
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    $\begingroup$ Why the votes to close? This seems a very reasonable question to me. (After the edits anyway.) $\endgroup$ – Todd Trimble Jan 19 '17 at 11:50
  • $\begingroup$ What about $f(x)=1/x$ (and $a=-1$)? $\endgroup$ – ACL Jan 20 '17 at 23:10
  • $\begingroup$ @ACL: OP does not ask about existence but about uniqueness. $\endgroup$ – András Bátkai Jan 21 '17 at 7:30

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.

  • $\begingroup$ Your comment is interesting, but I require all derivatives strictly monotone - it does not mean all derivatives non- negative. (e.g. ln(x) ). $\endgroup$ – H. Tomasz Grzybowski Jan 20 '17 at 10:35
  • $\begingroup$ Also, there is a question whether a smooth continuation is unique if we require all derivatives to be strictly monotonic. $\endgroup$ – H. Tomasz Grzybowski Jan 20 '17 at 10:49
  • $\begingroup$ exp(-x) is a better example of a smooth function whose all derivatives are strictly monotonic but some are negative, some positive. $\endgroup$ – H. Tomasz Grzybowski Jan 20 '17 at 14:38
  • $\begingroup$ @Tomasz Grzybowski: I edited my answer. $\endgroup$ – Alexandre Eremenko Jan 20 '17 at 19:23
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    $\begingroup$ There is also a Russian translation in Bernstein's Collected papers (also available on I-net) for those who do not read French. $\endgroup$ – Alexandre Eremenko Jan 20 '17 at 23:01

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