https://tetrationforum.org/hyperops_wiki/index.php?action=history&feed=atom&title=Totally_monotonicTotally monotonic - Revision history2026-08-10T16:39:01ZRevision history for this page on the wikiMediaWiki 1.35.8https://tetrationforum.org/hyperops_wiki/index.php?title=Totally_monotonic&diff=103&oldid=prevBo198214: initial draft2011-06-07T11:03:28Z<p>initial draft</p>
<p><b>New page</b></p><div>From <ref name="Szekeres1961">Szekeres, G. (1961). Fractional iteration of exponentially growing functions. J. Austral. Math. Soc., 2, 301–320.</ref>:<br />
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A function is said to be ''totally monotonic'' at $x_0$ if it has derivatives of any order and<br />
$$(-1)^{k+1} f^{(k)}(x_0) > 0 $$ for every $k>0$.<br />
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Denote by $\bf{M}$ the class of real functions $F(x)$ defined for $x>0$ and totally monotonic at every $x>0$; $\log(x)$ is a typical representative of $\bf{M}$. <br />
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'''Theorem 2'''. Let $A(x)$ denote the [[principal Abel function]] of $e^x-1$. Then $A(x)\in \bf{M}$ and $A(x)$ is the only Abel function of $e^x-1$ with this property.<br />
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== Forum Discussion ==<br />
http://math.eretrandre.org/tetrationforum/showthread.php?tid=37</div>Bo198214