# [FOM] Query on real functions

JoeShipman@aol.com JoeShipman at aol.com
Sun Oct 19 00:11:40 EDT 2003

```Here are some related and more precisely stated queries.

Consider the algebra of "absolutely monotone" real functions expressible as everywhere-convergent formal power series with nonnegative coefficients.  (For simplicity, restrict attention to [0, infinity) ignoring negative arguments).  This is closed under addition, multiplication, composition, differentiation, and integration.  Is there an absolutely monotone function f=a0 + a1x + a2x^2 + .... with the property that f(1)=2, f(n+1)=2^f(n) for n > 1?

A less precise followup: is there such an f which is in any sense "canonical" or naturally defined?

Related question: if f is absolutely monotone, what can be said about existence and uniqueness of absolutely monotone g such that g(g(x))=f(x) ?  If there is a "canonical" way to solve this functional equation, then we can get an analytic version of the "tower" function and much more.

-- JS

In a message dated 10/17/2003 3:10:53 PM Eastern Daylight Time, wwtx at pop.earthlink.net writes:

> At 5:19 PM -0400 10/14/03, JoeShipman at aol.com wrote:
> >Can anyone identify a real-valued function f continuous on [1,
> >infinity) with the following two properties:
> >
> >1) f eventually dominates any function in the sequence e^x, e^(e^x),
> >e^(e^(e^x)), ....
> >
> >2) f is defined in some other way than by defining it
> first at all
> >integers and then interpolating
> >
>
> If 2) could be replaced by some positive condition you need
> satisfied, it might help.
>
> Bill

```